i 3 Mathematica Fourier Fourier Dirac Dirac Fourier Haar 7.4 (MRA) 8 R L 2 {V j } j Z j Z V j V j+1 j Z V j = L 2 (R) f(2x) V j+1 f(

Transcription

1 Fourier 解析とウェーブレットの基礎 水谷正大著 大東文化大学経営研究所

2 i 3 Mathematica Fourier Fourier Dirac Dirac Fourier Haar 7.4 (MRA) 8 R L 2 {V j } j Z j Z V j V j+1 j Z V j = L 2 (R) f(2x) V j+1 f(x) V j V 0 ϕ(x) {T k ϕ(x)} V 0 V 0 = span{t k ϕ(x)} k Z ϕ V 0 L 2 (R) ϕ

3 ii D 2 j (x) = 2 j x(i Z) T k (x)x k(k Z) MRA V j W j j J V j = W j k V J, j > J, (8.8) k=1 L 2 (R) L 2 (R) = k Z W k (8.9) MRA 8.4 ϕ 8.2 {V j } j Z L 2 (R) {ψ} j,k Z L 2 (R) f V j ˆP j ˆP j f = ˆP j 1 f + f ψ j,k ψ j,k (8.7) k Z Fourier ϕ 6 7 Haar MRA Daubechies 9 Web [email protected]

4 iii I Fourier L 2 l L Fourier Fourier Dirac Dirichlet Fourier Fourier Gibbs Fourier Fourier Fourier Fourier

5 iv Fourier Fourier l 2 (Z N ) Fourier Fourier II Wavelet Haar Haar Haar Haar Haar Wavelet Haar Haar Haar Haar Haar MRA)

6 v 8.5 MRA Daubechies Fourier Daubechies Daubechies

7 I Fourier

8

9 V C n V x. x, y V inner product scalar product x y = n x k y k. (1.1) k=1 1.2 ( ) K V (1) v = 0 v v > 0. (1.2) (2) v, w V v w = w v. (1.3) x 1 x n

10 4 1 (3) a, b K u, v, w V u av + bw = a u w + b u w (1.4) antilinear av + bw u = a v u + b w u. (1.5) (1) 2 x, y V (1.1) x 2 interiour product n [ ] k=1 x k y k 1 x = x x = 0 i 1.13 x ket vector x bra vector P.A.M.Dirac *1 Dirac Schrödinger Ψ(x) Φ [11] x, y V x y y x x x x y x y x y K V K f f(a x + b y ) = af( x ) + bf( y ) V linear functional V V f, g V a K (f + g)( x ) = f( x ) + g( x ) (a f)( x ) = af( x ) V K V V dual space conjugate space *1 (bracket)

11 1.1 5 (1.1) x V x : y x y v V D v v u D v ( u ) = v u (1.6) x x x x = ( x t) [ ] = x 1 x 2... x n u a a u D(a u ) a u C n f (inner product space) Hilbert (pre Hilbert space) 1.3 ( ) V v V (norm) v =» v v (1.7) 2 L 2 2 l x, y V V x y = 1 4 x + y x y 2 + i 4 x + iy 2 i x iy 4

12 v w = 0 v = w v, w v(x) = w(x) {v k } v lim k v k v = 0 lim v? k = v. k 1.7 Mathematica x y x = (1 + i2, 2 + i4, 3 + i6) t, y = (8 + i4, 6 + i3, 4 + i2) t x = Table[i + I 2 i, {i, 1, 3}] y = Table[2 i + I i, {i, 4, 2, -1}] x.y 80 I y.x 80 I Conjugate[x].y I x.conjugate[y] I x x t Mathematica Transpose[ ] ConjugateTranspose[ ] x.y y.x 1.1 x y Conjugate[x].y Mathematica 1.2 L 2 l 2 2 Fourier f x t f (signal)

13 1.2 L 2 l 2 7 x f(x) t f(t) t R x f L 2 [a, b] {f(x) x [a, b]} 1.8 [a, b] 2 L 2 [a, b] L 2 [a, b] = f : [a, b] C b a f(x) 2 dµ(x) µ(x) dµ(x) = dx dµ(x) = w(x) dx b a f (x)f(x)dµ(x) < + weight density w(x) > 0 (1.8) 1.13 f, g L 2 f(x) (amplitude) f(x) L 2 [a, b] [a, b] 1.9 L 2 Lebesgue L 2 [a, b] dµ(x) = w(x)dx (, ) L 2 [, ] 1.16(2),(3) 1.10 ( 1 ) {f i (t)} n i=1 {c i} n i=1, c i C n i=1 c if i (t) = 0 c 1 =... = c n = 0 {f i (t)} i= ( ) N N N + 1

14 8 1 N N n n 1.12 span{1, x, x 2, x 3,...} L 2 [0, 1] 1/x L 2 [0, 1] 1 0 (1/x)2 dx = [ 1 1 x] = (L 2 ) f, g L 2 [a, b] L 2 f g L 2 f g L 2 = = b a b a f(x) g(x) dµ(x) (1.9) f(x) g(x) w(x)dx (1.10) µ(x) w(x) > 0 dµ(x) = w(x) dx 1.14 f g L = f f L 2 x [a, b] f(x) = 0 f f f g [a, b] L 2 f(x) = g(x)a.e [a, b] f(x) = g(x) C n x = (x 1,..., x n ) t y 1.1 f, g i- x i y i i x i y i f, g x- f(x) g(x) b a dxf (x)g(x)w(x) i b a dx w(x). f f f x f(x) 1.3

15 1.2 L 2 l { e i (x) } f(x) i=0,1,... f k (x) = k a i e i (x) i=1 (1) (1 x 2 ) d2 dx 2 P n(x) 2x d dx P n(x) + n(n + 1) = 0 (1.11) [ 1, 1] P n (x) Legendre P 0 (x) = 1, P 1 (x) = x, P 2 (x) = 3 2 x2 1 2, P 3(x) = 5 2 x3 3 2 x, P 4 (x) = 35 8 x x ,... {P n (x)} w(x) = 1 [ 1, 1] 1 1 (2) P m (x)p n (x) dx = 2 2n + 1 δ m,n. (1.12) x d2 dx 2 L n(x) (1 x) d dx L n(x) + nl n (x) = 0 (1.13) [0, ) L n (x) Laguerre L 0 (x) = 1, L 1 (x) = x + 1, L 2 (x) = x 2 4x + 2, L 3 (x) = x 3 + 9x 2 18x + 6, L 4 (x) = x 4 16x x 2 96x + 24,... {L n (x)} w(x) = e x [0, ] 0 L m (x)l n (x) e x dx = (n!) 2 δ m,n. (1.14)

16 10 1 (3) d 2 dx 2 H n(x) 2x d dx H n(x) + 2nH n (x) = 0 (1.15) (, ) H n (x) Hermite H 0 (x) = 1, H 1 (x) = 2x, H 2 (x) = 4x 2 2, H 3 (x) = 8x 3 12x, H 4 (x) = 16x 4 48x + 12,... {H n (x)} w(x) = e x2 [, ] H m (x)h n (x) e x2 dx = 2 n n! πδ m,n. (1.16) l (l 2 ) x = (..., x 2, x 1, x 0, x 1, x 2,... ) 2 l 2 { } l 2 = x x i 2 < + x i C. (1.17) i= 1.18 (l 2 ) x = (..., x 1, x 0, x 1,... ), y = (..., y 1, y 0, y 1,... ) l 2 l 2 x y l 2 x y l 2 = x ny n. (1.18) n= 1.19 (Hilbert ) = Hilbert 1.20 L 2 l 2 Hilbert

17 1.2 L 2 l (Schwartz ) V x, y V x, y x y, (1.19) x y x y = x y x y = x y e iϕ 0 e iϕ x ty 2 = e iϕ x ty e ıϕ x ty ( e = x 2 t iϕ x y + y e iϕ x ) + t 2 y 2 ( e = x 2 t iϕ x y + e iϕ x y ) + t 2 y 2 ( e = x 2 2tRe iϕ x y ) + t 2 y 2 = x 2 2t x y + t 2 y 2 t 2 2 D = 4 x, y 2 4 x 2 y ( ) V x, y V x + y x + y, (1.20) x y x = a y (a > 0)

18 12 1 Schwartz x + y 2 = x + y x + y = x 2 + 2Re ( x y ) + y 2 Schwartz x + 2 x y + y 2 = ( x + y ) 2 x y = x y V { v 1, v 2,..., v N } f V N f = c i v i. i=1 { v 1, v 2,..., v N } V (basis) V V = span{ v 1, v 2,..., v N } N V 1.24 V (1) x, y V (orthogonal) x y = 0 (2) V {e i } N i=1 (orthonormal) e i = 1 i j e i e j = 0 (3) V V 1 V 2 V 1 V 2 x V 1 y V 2 x y = V 1.54 Gram-Schmidt 1.26

19 V V 0 V { e 0 1, e 0 2,..., e 0 N } f V 0 N f = e 0 i f N e 0 i = e 0 i e 0 i f (1.21) i=1 i=1 { e 0 1, e 0 2,..., e 0 N } V 0 f V 0 N f = f i e 0 i. (1.22) i=1 f k f e k e 0 k f N = f i e 0 k e 0 N i = f i δ ik = f k. i=1 i= x 2 = y = z 3 (x, y, z)t R 3 (2, 1, 3) t {(x, y, z) 2x y + 3z = 0} 1.28 L 2 ([0, 1]) 2 f, g 0 x < 1 f(x) = 2 0 0, g(x) = 1 2 x < 1 f g = 1 f(x)g(x)dx = R 2 ϕ ψ 1 0 x < 1 ϕ(x) = 0, (1.23) 1 0 x < ψ(x) 1 2 x < 1 (1.24) 0 ϕ = ψ = 1 ϕ ψ = 0 ϕ Haar ψ Haar

20 Haar ϕ Haar ψ ϕ, ψ { ϕ, ψ } L 2 ϕ, ϕ L 2 = ψ, ψ L 2 = 1, ϕ, ψ L 2 = 0 Mathematical Mathematical ϕ ψ haarscaling[t_] := If[t < 0 1 <= t, 0, 1] haarwavelet[t_] := Piecewise[{{1, 0 <= t && t < 1/2}, {-1, 1/2 <= t && t < 1}}, 0] [, ] ϕ(x), ψ(x) [0, 1] Integrate[haarScaling[t] haarwavelet[t], {t, 0, 1}] Integrate[haarScaling[t]^2, {t, 0, 1}] Integrate[haarWavelet[t]^2, {t, 0, 1}] 1.31 [n, n + 1) 1 0 R I χ I χ [n,n+1) (x) Haar ϕ(x) χ [n,n+1) (x) = ϕ(x n) { ϕ(x n) } R n Z f(x) = χ [1/2,5/2) (x) f(x) = k= f n ϕ(x k) span{1, x, x 2, x 3,...} [a, b] Weierstrass 1.68 [a, b] i j x i x j = 1 0 xi x j dx = 1 i+j+1 { π,» 2 π cos nx } n=1,2... [0, π] Weierstrass 1.68 [0, π] f

21 f v(x) = f(cos 1 x) v(cos x) = f(x) v [ 1, 1] Weierstrass v(x) x f(x) cos x cos n x = 1 ñ Ç å Ç å ô n n 2 n 1 cos nx + cos(n 2)x + cos(n 4)x cos x n 1, cos s,..., cos ns 1.34 {» 2 π sin nx }n=1,2... [0, π] [0, π] f(0) = f(π) = 0 f x = 0, π 0 f(0) = f(π) = 0 [0, π] 1.35 π π sin x cos x dx = 0, π π sin 2 x dx = π π cos 2 x dx = π ß cos x 2, sin x L 2 ([ π, π]) π π ß cos n x, sin n x π π n=1,2,... (1.25) L 2 ([ π, π]) 1.36 (1.25) ß e inx 2π n Z (1.26) L 2 ([ π, π])

22 D = {(x, y) x 2 + y 2 1} 2 L 2 (D) = ß f : D C f(x, y) 2 dxdy < + D L 2 (D) f g := D f(x, y) g(x, y) dxdy. {z n (x, y) = (x + iy) n } n=0,1,2,... L 2 (D) V V 0 f V f 0 f V 0 orthogonal projection ˆ P V0 f f V f f 0 V 0 (1.27) f 0 = P V0 f f = f 0 + w ( f 0 V 0, w V 0 ) (1.28) f V V 0 P ˆ V0 f f = f 01 + w 1 = f 02 + w 2 f 0i V 0, w i V 0, (i = 1, 2) f 01 f 02 = w 2 w 1. f 01 f 02 f 01 f 02 2 = f 01 f 02 w 2 w 1 = 0. f 01 = f 02 w 1 = w V V 0 { e 0 1,..., e 0 N } f V N N f 0 = e 0 j f e 0 j = e 0 j e 0 j f j=1 j=1

23 f 0 = ˆP 0 = P ˆ V0 f P ˆ V0 N e j e j (1.29) j=1 V 0 f 0 V 0 f f0 e 0 k = f e 0 k f0 e 0 N k = f e 0 k f e 0 j e 0 j e 0 k j=1 = f e 0 k f e 0 k = 0 (k = 1,..., N) (f f 0 ) V ˆP = ˆP 2 = ˆP. n e i e i i=1 ( n ) ( n ) n n ˆP 2 = e i e i e k e k = e i e i e k e k i=1 k=1 i=1 k=1 e i e k = δ ik n = e i e i = ˆP. i=1 (1.21) 1.26 Dirac 1.41 V V 0 { e 0 j }N k=1 N e 0 k e0 k ˆP V0 = N e 0 k e 0 k (1.30) k=1

24 18 1 V V 0 ˆP V0 : V V 0 V f V 0 f = ˆP V0 f f = N e 0 k e 0 k f (1.31) k=1 { e 0 k f } { e0 k } f ( e 0 1 f,..., en 0 f )t Hilbert 1.42 ( ) V 0 Hilbert V f V f V 0 f 0 = ˆP V0 f f f = f 0 + w ( f 0 V 0, w V 0 ) ˆP V0 f f V 0 δ = inf f g g V 0 inf V 0 f n V 0, f f n δ (n ) { f n } { f n } f 0 = lim n f n P V0 f f n V 0 V 0 lim n f n V 0. f f n f f 0 = δ f f 0 = min g V 0 f g w V 0 t f f 0 + tw V 0 2 l(t) = f (f 0 + tw) l(t) = f f 0 tw 2 = f f 0 2 2tRe ( f f 0 w ) + t 2 w

25 f 0 f t = 0 l(t) l (0) = 0 Re ( f f 0 w ) f f 0 w = 0 ( w V 0 ) f f 0 w V 0 f 0 = P V0 f { f n } { f n } Cauchy (f f n ) + (f f m ) 2 + f m f n 2 = 2 f f n f f m 2 4 f f n + f m f m f n 2 = 2 f f n f f m 2. ( f n + f m )/2 V 0 f fn+fm 2 δ 2 f m f n 2 2 f f n f f m 2 4δ 2 2 n, m 2δ 2 +2δ 2 4δ 2 = 0 f n f m 0 (n, m ) ß 1.43 L 2 ([ π, π]) V 0 e 0 1(x) = cos x, e 0 π 2(x) = sin x π f(x) = x V 0 f 0 (x) 1.41 f 0 (x) = e 0 1 f e 1 (x) + e 0 2 f e2 (x) 1 [ π, π] 0 2 e 0 2 f = 1 π π π x sin x dx = 2 π f 0 (x) = 2 sin x Mathmematica 1.43

26 20 1 e1[t_] := Cos[t]/Sqrt[Pi]; e2[t_] := Sin[t]/Sqrt[Pi]; ip[p1_, p2_] := Integrate[p1 p2, {t, -Pi, Pi}] (* t p1 p2 *) Projection[t, e1[t], ip] Projection[t, e2[t], ip] 1.45 (1.23) Haar ϕ(x) (1.24) Haar ψ(x) L 2 [0, 1] V 1 = span { ϕ(x), ψ(x) } f(x) = t ϕ f = 1 0 x dx = 1 2, ψ f = 1/2 f V 1 f 1 (x) 0 1 x dx x dx = 1 1/2 4 f 1 (x) = ϕ f ϕ(x) + ψ f ψ(x) = ϕ(x)/2 ψ(x)/4 1/4 0 x < 1/2 = 3/4 1/2 x < V n = ß 1 span, cos k x, sin k x 2π π π k=1,...,n L 2 ([ π, π]) f(x) = x 2 n = 1, 2, 3 V 1, V 2 V 3 V 1.47 Mathematica 1.46 f(x) = x 2 V 1, V 2 V 3 Mathematica 1.44 ip f g g(t) f(t) g(t) Projection[f[t], g[t], ip] Mathematica Projection 1.1 f(x) = x 2 L 2 ([ π, π]) V 3 π 2 /3 4 cos x + cos 2x 4/9 cos 3x ([ π, π] x 2 V 1, V 2

27 V n=3 f(x) = x 2 n f(x) base[t_, n_] := Flatten[{1/Sqrt[2 Pi], Table[{Cos[k t]/sqrt[pi], Sin[k t]/sqrt[pi]}, {k, 1, n}]}] (* *) ip[p1_, p2_] := Integrate[p1 p2, {t, -Pi, Pi}] (* *) f[t_] = t; (* t *) pf1[t_] = Sum[Projection[f[t], base[t, 1][[i]], ip], {i, 1, Length[base[t, 1]]}] (*n=1 *) Plot[pf1[t], {t, -Pi, Pi}] (* *) pf2[t_] = Sum[Projection[f[t], base[t, 2][[i]], ip], {i, 1, Length[base[t, 2]]}] (*n=2 *) Plot[pf2[t], {t, -Pi, Pi}] (* *) pf3[t_] = Sum[Projection[f[t], base[t, 3][[i]], ip], {i, 1, Length[base[t, 3]]}] (*n=3 *) Plot[pf3[t], {t, -Pi, Pi}] (* *) f2[t_] = t^2; (* t^2 *) pf23[t_] = Sum[Projection[f2[t], base[t, 3][[i]], ip], {i, 1, Length[base[t, 3]]}] (*n=3 *) Plot[pf23[t], {t, -Pi, Pi}] Haar ϕ(x) Haar ψ(x) L 2 ([0, 1]) span {ϕ(x), ψ(x), ψ(2x), ψ(2x 1), ψ(2x + 1)} f(x) = x

28 Mathematica ip iph 1.2 Haar ϕ(x) Haar ψ(x) {ϕ(x), ψ(x), ψ(2x), ψ(2x 1)} L 2 ([0, 1]) f(x) = x x Haar haarscaling[t_] := If[t < 0 1 <= t, 0, 1] (*Haar scaling *) (*Haar wavelet*) haarwavelet[t_] := Piecewise[{{1, 0 <= t && t < 1/2}, {-1, 1/2 <= t && t < 1}}, 0] (* *) hbasis[t_] = {haarscaling[t], haarwavelet[t], haarwavelet[2t], haarwavelet[2t-1]} f[t] = t; (* t *) iph[p1_, p2_] := Integrate[p1 p2, {t, 0, 1}] (* *) (* *) hpf[t_] = Sum[Projection[f[t], hbasis[t][[i]], iph], {i, 1, 4}] Plot[hpf[t], {t, 0, 1}] f(x) = sin 2πx hbasis[t] Haar ϕ(x) Haar ψ(x) Mathematica

29 haarbasis[t_, n_] := Flatten[{haarScaling[t], Table[haarWavelet[2^j t - k], {j, 0, n}, {k, 0, 2^j - 1}]}] H n, (n 1) ß { } H n = span ϕ(x), ψ(2 j x k) j=0,...,n,k=0,...,2 j 1 (1.32) 1.3 f(x) = sin 2πx (1.32) Haar H n = 1 f[t_] := Sin[2Pi t]; haarbasis[t, 1]; hpf3[t_] = Sum[Projection[f[t], haarbasis[t, 1][[i]], iph], {i, 1, Length[haarbasis[t, 1]]}]; haarbasis[t, 3] H 3 f(x) = sin 2πx H n f(x) H n f(x) n = 4 H 4

30 ( ) V 0 V V 0 (orthogonal complement) V 0 V 0 = { v V w V 0 w v = 0} 1.52 V 0 V f V f 0 V 0 f 1 V 0 f = f 0 + f 1 V V = V 0 V 0 f V f 0 f V 0 f 1 = f f f 1 V 0 f = f 0 + (f f 0 ) = f 0 + f 1 f 1 V 0. f 1 V V 0 = {(x, y, z) t 2x y + 3z = 0} { e1 = 1 21 (1, 4, 2)t, e 2 = 1 6 (2, 1, 1)t} e 1 = e 2, e 1 e 2 = 0 V 0 v = (x, y, z) t R 3 V 0 v 0 v 0 = e 1 v e 1 + e 2 v e 2 Å ã Å ã x 4y 2z 2x + y z = (1, 4, 2) t + (2, 1, 1) t e 3 = 1 14 (2, 1, 3)t e 3 = 1 e 3 V 0 e 3 e 1 = e 3 e 2 = 0 v 1 e 3 = e 3 v e 3 2x y + 3z = (2, 1, 3) t 14 v 1 V 0 v 1 e 1 = v 1 e 2 = 0

31 x-y x -y 1.54 (Gram-Schmidt ) V N { v 1, v 2,..., v N } V { e 1, e 2,...} e j { v 1, v 2,...}» e 1 = ẽ 1 / ẽ 1, ẽ 1 = v 1» e 2 = ẽ 2 / ẽ 2, ẽ 2 = v 2 e 1 v 2 e 1...» e N = e N / e N,... N 1 e N = v N e k v N e k k= L 2 ([0, 1]) {x n } n=0 Gram- Schmidt [0, 1] Mathematica Mathematica {1, x, x 2, x 3 } L 2 ([0, 1]) Orthogonalize[{1, t, t^2, t^3}, Integrate[#1 #2, {t, 0, 1}] &] 1.56 [ 1, 1] {1, x, x 2, x 3,...} Gram-Schmidt {» } n P n(x) n=0,1,2,... P n (x) [ 1, 1] Legendre 1.16(1) P n (x) Legendre (1 x 2 ) d2 P n (x) dt 2 2x dp n(x) dt + n(n + 1)P n (x) = 0

32 26 1 Legendre [ 1, 1] P 0 (x) = 1, P 1 (x) = x, P 2 (x) = 1 2 (3x2 1), P 3 (x) = 1 2 (5x3 3x), P 4 (x) = 1 8 (35x4 30x 2 + 3) Mathematica Orthogonalize[{1, t, t^2, t^3}, Integrate[#1 #2, {t, -1, 1}] &] 1.57 Legendre P n (x) Rodrigues P n (x) = 1 d n (x 2 1) n 2 n n! dx n P n (x) = 1 2 n n k=0 Ç å 2 n (x 1) n k (x + 1) k k x = 1 P n (x) {P n (x)} n=0,1,2,... Mathematica P 0 (x), P 1 (x),..., P 20 (x) [0, 1] 1.4 Legendre P n(x) [ 1, 1] n = 0, 1, 2,..., 20

33 V V 0 V E 0 = { e 0 1, e 0 2,..., e 0 N } f V {f n } N n=1 f n = e 0 n f f E 0 Fourier Fourier-type expanding coefficients v f V 0 N e 0 n f e 0 k k=1 f N f n e 0 k 2 = k=1 k=1 ((x) fn e 0 k(x) ) 2 dx 0 N N N N 0 f f n e 0 k 2 = f 2 + f k 2 f i e 0 i f f j f e 0 j k=1 k=1 N N = f 2 + f k 2 2 f i 2 i=1 V V 0 V E 0 f V N f k 2 f 2 (1.33) k=1 (1.33) Bessel N N k=1 f k 2 f 2 i=1 j=1

34 V piecewise continuous *2 f V V E = { e n } n=1,2,... f f n e n ε > 0 n f f k e k 2 < ε, f k = e k f, n > N k=1 N f L 2 V complete total V Bessel 1.60 ( ) V E = { e n } n=1,2,... f V {f n = e n f } Bessel f 2 = k f n 2 Bessel g V {g n = e n g } Parseval f g = k f k g k 2.45 = [5] 2 f = k f ke k *2 1 piecewise smooth

35 1.5 L Hilbert H { e k } k=1,2,... e k e k Î H = k e k e k (1.34) H H f f = ÎV f f = k e k e k f { e k f } { e k } 1.5 L 2 {f n } f L ( ) {f n } f [a, b] (pointwise convergence) x [a, b] ε > 0 N = N(ε, x) n N f n (x) f(x) < ε N ε x x n N f n (x) f(x) 1.63 ( ) {f n } f [a, b] (uniform convergence) ε > 0 x [a, b] N = N(ε)

36 30 1 n N f n (x) f(x) < ε N ε n N [a, b] f n (x) f(x) 1.64 (L 2 ) {f n } f L 2 [a, b] ε > 0 N = N(ε) n N f n (x) f(x) L 2 < ε n N [a, b] t f n (x) f(x) L 2 (convergence in mean) 1.65 OK L 2 L 2 L {f n (x) = x n } n=1,2,... f(x) = 0 [0, 1) 0 x < 1 x n 0, (n ). {x n } [0, 1) x 1 x n ε = x n < ε x = 0.5 n 10 x = 0.9 n 66 ε x N(ε, x) r < 1 [0, r] {x n } [0, r] 0 r N > ε > 0 ε n N [0, r] f n (x) < ε {x n } [0, 1] L 2 1 f n 2 L = (x n ) 2 1 dx = 0, 2 2n (n ).

37 1.5 L {f n } [a, b] f {f n } f L 2 [a, b] {f n } f L 2 [a, b] x [a, b] ε > 0 N = N(ε) n N f n (x) f(x) < ε b f n f 2 L = f 2 n (x) f(x) 2 dx a b {f n } f L 2 [a, b] a ε 2 dx = ε 2 (b a) [0, 1] {f n } f n (x) = 1 0 < x 1 n 0 {f n } [0, 1] f(x) = 0 L f n (x) f(x) 2 dx = 1 n 0 dx = 1 0(n ). n x [a, b] 1 > ε > 0 n N f n (x) 0 < ε N n f n (x) 0 = 1 [a, b] f(x) ( (x a)(1 δ) + (b x)δ ) /(b a) x [δ, 1 δ] [0, 1], δ < 1 2 [0, δ), (1 δ, 1] f [0, 1] 1.68 [Weierstrass ] [0, 1] f 2n, n = 1, 2,... p n (x) = 1 2I n 1 0 f(t) ( 1 (t x) 2) n dt, In = 1 0 (1 t 2 ) n dt

38 32 1 ε > 0 n = n(ε) f(x) p n (x) < ε, x [0, 1] V W T : V W (linear) T (c 1 v + c 2 w ) = c 1 T v + c 2 T w, v i V, c i C V V 0 {e k } 1.42 P : V V 0 v 1, v 2 V P (a v 1 + b v 2 ) = ap (v 1 ) + bp (v 2 ) 1.71 ( ) { v 1,..., v n } V { w 1,..., w m } W T : V W v V x = (x 1,..., x n ) t w W y = (y 1,..., y m ) t T A T a ij = w i T v j y = A T x y 1 a a 1n x y i = a j1... a jn x j y m a m1... a mn x n T m m T = a ij w i v j (1.35) i=1 j=1 A T = (a ij ) { v j } { w i } T

39 v = n j=1 x j v k V T w = m k=1 y k w k W ( n n ) w = y k w k = T x j v j k=1 = j=1 n x j T v j. j=1 w i w i w = y i y i = = n w i x j T v j j=1 n a ij x j j=1 T 1.42 V I W W I W T = I W T I v = = m i=1 j=1 m w i w i T i=1 n w i w i T v j v j n v j v j j= T : V W T T = { max T v } v V v = 1 T (bounded operator) 1.73

40 V W T : V W (adjoint) w T v W = T w v V T : W V v V w W w u V W w u : V W ( w u ) = u w u w : W V 1.76 w u : V W u w : W V (1.35) ( m n T = w i w i T v j v j i=1 j=1 n m = v i w j T v i w j i=1 j=1 i=1 j=1 ) T A T V W { v i } { w j } T A T = (a ij ) = ( w i T v j ) v V w W x y m n y A T x W = yi a ij x j = ( n m ) a ijy i x j = j=1 i=1 j=1 i=1 = A T y x V ( n m a ji i) y x j

41 a ij = a ji T T A T T A T A T 1.78 g [a, b] T g L 2 ([a, b]) : L 2 ([a, b]) T g f(x) = g(x)f(x). T g T g h(x) = g(x) h(x) h T g f [a,b] = b h(x) g(x)f(x) dx = b a a (g(x) h(x)) f(x) dx = g h f [a,b] = T g h f [a,b] ( ) T 1 : V W T 2 : W U 2 (T 2 T 1 ) = T 1 T 2 v V u U u T 2 T 1 v U = T 2 u T 1v W = T 1 T 2 u v V u T 2 T 1 v U = (T 2 T 1 ) u v V ( ) V V 0 P P 0 = P 0

42 36 1 V 0 { e 0 j }N j=1 (1.29) P 0 = N e 0 k e 0 k k=1 ξ η ( ξ η ) = η ξ P 0 = P 0 [ ] 1.52 v, u V v 0, u 0 V 0 v 1, u 1 V 0 v = v 0 + v 1 u u 0 + u 1 P 0 v = v 0, P 0 u = u 0 v P 0 u = v 0 + v 1 u 0 = v 0 u 0. P 0 v u = u 0 u 0 + u 1 = v 0 u 0. P 0 v, u = v, P 0 u P 0 = P (Hermite ) H H H = H (self-adjoint) Hermite (hermitian) 1.82 Hermite

43 37 2 Fourier 2.1 Fourier m k 2.1 x F = kx m d2 x dx 2 = kx ω 2 = k m d 2 x dx 2 = ω2 x (harmonic ocillator) 2.1 k x F = kx

44 38 2 Fourier x(t) = A sin(ωt + δ) x(t) = a cos(ωt + γ) A (amplitude) δ, γ t = 0 (phase) 1 T [sec] (period) x(t) = x(t + T ) T = 2π ω ν = 1/T [Hz] (frequency) π (angular frequency) ω = 2πν [rad/sec] k U(x) U(x) = k 2 x2 = mω2 2 x2 = m 2 ω2 A 2 sin 2 (ωt + δ). K K = m Å ã dx 2 = m 2 dt 2 ω2 A 2 cos 2 (ωt + δ). E E = K + U = mω2 2 A2 E 2 f(t) Fourier ˆf(λ) λ f(t) ˆf(λ) 2 (power spectrum ) 2 (traveling wave) 1 λ [m] wavelength v [m/sec] v = λ ν

45 sin ωt C Hz sin 3 2 ωt E sin 3ωt G CEG 1 : 3 2 : 3 (consonance) : 2 : : 3 : f 0 (t) L 2 ([ π, π]) f 0 (t) = 3 sin 2t + 4 sin 3t + 2 sin 6t 3, 4, 2 2 CEG 2 : 3 : 6 [ π, π] f 0 2.3(a) f 1 f (b) f 1 f 1 (t) = sin 2t + 4 sin 3t + 2 sin 6t cos 100t 2 f 0 f 1 Fourier f 0 f 1 Fourie

46 40 2 Fourier (a) (b) 2.3 (a) f 0(t) = 3 sin 2t+4 sin 3t+2 sin 6t (b) f 1(t) = f 0(t)+0.2 cos 100t f cos 100t f 0 f 1 Fourier 0.1 f 0 (noise) f 1 f 0 (low pass filter) f 1 f 0 L 2 = f 0 L 2 f 1 L 2 = L 2 f 1 Fourier 0.2 cos 100t f 0 f Mathematica f 0 f 1 L 2 f 1

47 2.2 Fourier 41 nf1=sqrt[integrate[(f0[t] Cos[100 t])^2, {t, -Pi, Pi}]] T v ω 20Hz 15KHz 20KHz 300Hz 3.4KHz CD 22,050Hz CD Fourier Fourier 2.2 (Fourier Fourier ) [ π, π) f(x) a n = a n (f) = 1 π π π f(x) cos nx dx (2.1) b n = b n (f) = 1 π π π n Fourier n 0 f(x) sin nx dx (2.2) S[f] = a (a n cos nx + b n sin nx) (2.3) n=1 f Fourier Fourier 2π [ a, a)

48 42 2 Fourier Fourier f Fourier S[f] f(t) Fourier Fourier Fourier f Lebesgue [a, b] a x 1 < x 2 < < x N b x1 ϵ x2 ϵ x3 ϵ lim ϵ 0 a x 1+ϵ x 2+ϵ b x N +ϵ f(x) dx < + Riemann f Riemann Lebesgue Fourier 2.3 ( Fourier ) [ π, π) f(x), f(x) Fourier 1 2π n= Fourier 2.4 f n = 1 2π π f n e inx, (2.4) π f(x)e inx dx (2.5) f n = a n ib n, (n = 0, ±1, ±2,... ) 2 f n e inx + f n e inx = a n cos nx + b n sin nx f a n, b n (2.4) (2.3) a n = a n, b n = b n f n = f n (2.6)

49 2.2 Fourier (2.6) Fourier 2.6 (Fourier ) (1) Fourier ß 1 2π, 1 π cos nx cos mx dx = π π 1 π π π 1 π π π (2) Fourier sin nx sin mx dx = cos nx sin mx dx = 0. cos nx sin nx, L 2 ([ π, π]) π π n N ß 1 2π e inx n Z 1 n = m 0 2 n = m = 0, 0 m n 0 m n 1 n = m 0, L 2 ([ π, π]) 1 π e inx e imx = δ nm. 2π π f(x) = α α n cos nx + β n sin nx (2.7) n=1 f(x) cos mx 1 π π f(x) cos mx dx π π = α 0 2π = α m 1 π π π π cos mx dx + n=1 f(x) sin mx dx = β m Å αn π π π cos nx cos mx dx + β n π π π ã sin nx cos mx dx

50 44 2 Fourier (2.7) Fourier (2.3) n= γ ne inx f Fourier f Fourier f Fouerie Fouerie L Dirac Dirac V 1, 2,..., n { j } j=1,...,n k j = δ k,j V 1.61 (1.34) V ÎV n Î V = j j j=1 (2.8) V f I V n f = ÎV f = j j f (2.9) j=1 V f n f j f f j

51 2.3 Dirac n 1.61 [ π π] Fourier einx 2π π π Å e imx 2π ã e inx 2π dx = δ m,n Dirac n x n = einx 2π (2.10) x ˆx x ˆx x = x x (1.3) m x = x m (2.11) π π m x x n dx = δ m,n = m n (2.12) Fourier (2.8) n [ π, π] Î Î = n n (2.8 ) n= (2.9) f = Î f = n= j j f (2.9 )

52 46 2 Fourier (2.10) x f = x Î f = n= x n n f = n= e inx 2π n f (2.13) (2.4) n f = f n Fourier Fourier f n = n f Dirac f n = π π e inx π e inx π f(x) dx = x f dx = n x x f dx = n f 2π π 2π π (2.14) x n f (2.14) m n (2.12) [ π, π] f Î x Î = π π x x dx (2.15) (2.8 ) x x x x (2.14) n f π n f = n Î f = n x x f dx = π π π e inx π e inx x f dx = f(x) dx 2π π 2π (2.16) y (2.15) y = Î y π y = Î y = x x y dx (2.17) π

53 2.3 Dirac 47 y x f y y f (2.15) π y f = y Î f = y x x y dx = π π π y x f(x) dx (2.18) (2.17) x y (2.18) y x (2.15) x Dirac x y = y x δ(x y) δ(x)?? Dirac (2.18) Dirac (, ) f(y) = δ(x y)f(x) dx (2.18 ) Dirac functional δ(x y) f(x) f(y) Dirac x ˆx x x ˆx x x x y, x(x y) y x y x = 0 x = 1 δ(x y) = y x = x y [11] δ(x y) = 0 if x y if x = y, δ(x y) dx = 1 x y = δ(x y) f x f (2.15) x f = x Î f = x x x f dx = δ(x x )f(x ) dx = f(x) f(x) f x f x x f x f(x) f f(x)

54 48 2 Fourier Dirac f(x) Fourier 2.3 [ π, π] { x n } n Z f(x) Fourier f(x) = x f = x Î f = x n n f = = = n= n= n= e inx 2π n Î f = π π n= n= e inx 2π n f = e inx π 2π π n= f n e inx 2π n y y f dynonumber (2.19) e inx 2π f(y) e iny 2π dy (2.20) ( π ) e in(x y) f(x) = f(y) dy (2.21) 2π π n= (2.20) 2π [ π, π] f(x + 2πi) = π π ( n= e in(x y) 2π ) f(y) dy (2.22) f(y) f(x + 2πi) n= n= e in(x y) 2π = k= [ e inx = π 2π δ(x y 2πk) (2.23) ] cos nx = n=1 δ(x 2πk) (2.24) k=

55 2.3 Dirac 49 Dirac Dira comb 2πk f(x) 2a f(x) = f(x + 2a) f(at/π) = f(at/π + 2a) = f(a(t + 2π)/π) g(t) = f(at/π) t [ π, π) 2π g(t) Fourier S[g] = a a k cos kt + b k sin kt k=1 t = πx/a 2.9 ( [ a, a) Fourier ) f(x) [ a, a) f Fourier S[f] = a a k cos πk a x + b k sin πk x, (2.25) a Fourier a k = 1 a a a k=1 f(x) cos πk x dx, k 0 (2.26) a b k = 1 a a a f(x) sin πk x dx (2.27) a Fourier 2a (2.25) f(x) f n = a n= a f n e iπnx/a 2a, (2.28) f(x) e iπnx/a 2a dx (2.29)

56 50 2 Fourier [0, a] f(x) [ a, a) (even extension)f ev (odd extension)f od 2.4 [0, a] f(x) [ a, a) f ev (x) f ev (x) = f(x) 0 x < a f( x) a x < 0 [0, a] f(x) [ a, a) f od (x) f od (x) = f(x) 0 x < a f( x) a x < 0 (a) (b) 2.4 [0, a) f(x) (a) (b) 2.10 [0, a) f(x) (1) [ a, a) f ev (x) [ a, a) Fourier S[f ev ] = a a k = 2 a k=1 a 0 a k cos kπx a, f(x) cos kπx a dx, k 0.

57 2.3 Dirac 51 (2) [ a, a) f od (x) [ a, a) Fourier S[f od ] = b k sin kπx a, k=1 a b k = 2 a f(x) sin kπx a [a, b) f(b) = f(a) R b a 2.5 (periodic extension) b a dx 2.5 [a, b) f(x) [0, a] [ a, a) [ a, a) [ a, a) 2a R f(x) = f(x + 2a) (2.26) (2.27) Fourier 2a 2.12 F (x) 2a c a a+c F (x) dx = a a+c F (x) dx F 2a x x 2a a+c F (x) dx = a+c a a F (x ) dx

58 52 2 Fourier a+c a+c F (x) dx = = a+c a a a a+c a F (x) dx F (x) dx = a+c a a + F (x) dx a+c Fourier f g 2a f(x) g(x) Fourier (2.28) (2.29) f(x) n= e iπnx/a a f n, f n = f(x) e iπnx/a dx 2a a 2a 2.13 ( ) 2a f g (convolution) f g (f g)(x) = a a f(x y)g(y) dy = a a f(y)g(x y) dy (2.30) 1 2a R ) 2.14 (Fourier ) f g 2a α, β C (i) (αf + βg) n = αf n + βg n, (ii) (f g) n = 2af n g n, (iii) f n f 1, f 1 = 1 a 2a a f(x) dx.

59 2.4 Dirichlet 53 (i) Fourier (ii) (f g) n = 1 2a = 1 2a = 2a (iii) a a a (f g)(x)e iπnx/a dx Å a a a a 1 2a a ã f(x y)e iπn(x y)/a g(y)e iπny/a dy dx g(y)e iπny/a dy 1 a f(x y)e iπn(x y)/a dx = 2ag n f n. 2a a f n 1 a f(x)e iπnx/a dx = 1 a f(x) dx. 2a a 2a a 2.4 Dirichlet Fourier Dirichlet (Dirichlet kernel) 2.15 (Dirichlet ) T Dirichlet D N (x, T ) = sin (2N + 1) π ) T x T sin π T x (2.31) T = 2π 2.6 Å sin N + 1 ã x 2 D N (x) = 2π sin x 2 (2.32) 2.16 (Dirichlet ) ( ) D N (x) = 1 1 N π 2 + cos nx n=1 = 1 2π N e inx. (2.33) n= N

60 54 2 Fourier 2.6 T = 2π, N = 5 Dirichlet D N (x) Euler e ikx = cos kx + i sin kx N n= N z C N n=0 N N e inx = 1 + (e inx + e inx ) = cos nx z n = 1 zn+1 1 z n= N n=1 z n = 1 + z 2zN+1 1 z n=1 z = e ix e ix/ N e inx = cos x 2 [ cos ( ) ( ) ] N x + i sin N x i sin x. 2 n=1

61 Dirichlet D N (x) D N (x) = cos 2 x cos ( ) N x sin x 2 = 2 sign( ) x C 2.18 sign(z) = D N (x, T ) = 1 T 2.19 T/2 T/2 z/ z (z 0) 0 (z = 0). N n= N e i 2πn T x = 1 T ( N sin nx = n= N n=1 N n= N ) cos 2πn T x i sign(n)e inx D N (x, T ) dx = 1 (2.34) 2.5 Fourier Dirichlet D N (x) (sine integral) Si(x) Si(x) = x 0 sin t t dt, 0 x < = x x x5 600 x x Si(x) 2.7

62 56 2 Fourier 2.7 Si(x) 2.20 Si(x) sin t t 2.21 Mathematica SinIntegral[ ] Series[f[x], {x, a, n}] f(x x = a (x a) n 2.22 ( ) (i) Si(x) π, 3π, 5π,... 2π, 4π, 6π,... (ii) Si(0) = 0. (iii) lim x Si(x) Si( ) = π 2 = (iv) 0 x Si(x) Si(π) = (i) d Si(x)/dx = sin x/x = 0 x (ii) (iii) 0 < a < b < + b sin t ï dt t = cos t ò b b cos t t a t 2 1 a + 1 b b + dt a t 2 = 1 a + 1 b 1 b + 1 a = 2 a a a a 0 Si( ) Cauchy (n+1)π nπ sin t t dt = π 0 sin t t + nπ dt > 1 (n + 1)π π 0 sin t dt = 2 (n + 1)π > 2 n+2 dt π n+1 t

63 nπ 0 sin t t dt > 2 π n+1 1 dt t = 2 log(n + 1) +, π n 2.28 d(x) = 1 sin x 2 x = 0 d(0) = 0 [0, π] 2 x Riemann-Lebesgue 2.32 π lim n 0 Å 1 sin(t/2) 2 ã sin(n + 1 )t dt = 0. t 2 (n+ 1 2 )π sin t π sin(n Si( ) = lim dt = lim )t dt n 0 t n 0 t Riemann-Lebesgue 1 = lim n 2 π = lim n 2 π 0 π 0 sin(n )t sin t 2 D n (t) dt = π 2 dt D n (x) Dirichlet (2.34) [Si( ) ] Fourier 3.4 [ 1, 1] 1 0 f(x) = χ [ 1,1] (x) f(x) = 1 1 x 1 0 f(x) f Fourier f(λ) f(λ) = 1 2π = 2 sin λ π λ f(t)e iλt dt = 1 2π f(t) cos λt dt

64 58 2 Fourier Fourier f(x) = 1 2π f(λ)e iλx dλ f(0) = 1 = 1 2 sin λ 2π π λ dλ, sin λ λ dλ = 2 Si( ) = π. (iv) 0 2kπ < t < (2k + 1)π sin t/t > 0 0 < (2k + 1)π < t < (2k + 2)π sin t/t < 0 Si ((n + 1)π) Si(nπ) = (n+1)π nπ sin t t dt, n 0 n n 2.7) (i) 0 x Si(x) Si(π) Si(π) 2.6 Fourier Fourier f Fourier [0, π) x 0 x < π/2 f(x) = π x π/2 x < π (2.35) [ π, π) C 1 Fourier

65 2.6 Fourier 59 [ π, π) f ev (x) = f(x) 0 x < π f( x) π x < [0, π) [ π, π) 2.10 f ev [ π, π) Fourier Fourier f(x) a 0 = 2 π a j = 2 π = π 0 π 0 f(x) dx = π 2 f(x) cos jx dx = 2 π π 2 4 cos jπ/2 2 cos jπ 2 πj 2, j 1 0 x cos jx dx + 2 π π π 2 (π x) cos jx dx a 0 Fourier a 4k+2 2 a 4k+2 = π(2k + 1) 2, k 0, Fourier S[f ev ] = a π k=0 1 cos(4k + 2)x. (2k + 1) 2

66 60 2 Fourier f ev N S N (x) S 2 (x) = π 4 S 6 (x) = π 4 S 10 (x) = π 4 S 14 (x) = π 4 2 cos 2x π 2 cos 2x π 2 cos 2x π 2 cos 2x π 2 cos 6x 9π 2 cos 6x 2 cos 10x 9π 25π 2 cos 6x 2 cos 10x 9π 25π 2 cos 14x 49π [ π, π) [0, π) f(x) [ π, π] f ev 2,6,10,14 k = 0, 1, 2, 3 Fourier S N (x) f ev [ π, π) 2π [ π, π) f od (x) = f(x) 0 x < π f( x) π x <

67 2.6 Fourier [0, π) [ π, π) 2.10 f od [ π, π) Fourier Fourier f(x) π b j = 2 4 sin jπ/2 f(x) sin jπ = π 0 πj 2 0 j = 2m, = 4( 1) m π(2m + 1) 2 j = 2m + 1, m = 0, 1, 2,.... Fourier S[f od ] = m=0 4( 1) m sin(2m + 1)x. π(2m + 1) 2 f od N S N (x) S 1 (x) = 4 sin x, π S 3 (x) = 4 sin x π S 5 (x) = 4 sin x π S 7 (x) = 4 sin x π 4 sin 3x, 9π 4 sin 3x 4 sin 5x + 9π 25π, 4 sin 3x 4 sin 5x + 9π 25π 4 sin 7x 49π. [ π, π) 2.11

68 62 2 Fourier 2.11 [0, π) f(x) [ π, π] f od 1,3,5,7 Fourier S N (x) f od [ π, π) 2π 2.23 Fourier Mathematica Fourier [0, π) f[x_] := Piecewise[{{x, 0 <= x && x <= Pi/2}, {Pi - x, Pi/2 < x <= Pi}}] fe[x_] := Piecewise[{{x+Pi, -Pi <= x && x < -Pi/2}, {-x, -Pi/2 < x && x < 0}, {x, 0 <= x && x <= Pi/2}, {Pi-x, Pi/2 < x <= Pi}}] fo[x_] := Piecewise[{{-x-Pi, -Pi < +x && x < -Pi/2}, {x, -Pi/2 <= x && x <= Pi/2}, {Pi-x, Pi/2 < x <= Pi}}] n Fourier n Fourier fc[x_, n_] := FourierCosSeries[f[x], x, n] fs[x_, n_] := FourierSinSeries[f[x], x, n]

69 2.7 Fourier ,6,10,14 k = 0, 1, 2, 3 Fourier [ π, π] Plot[{fc2[x], fc6[x], fc10[x], fc14[x]}, {x, -Pi, Pi}] 2.7 Fourier Fourier Haar Scaling Fourier 1.48 Haar scaling [ 2, 2) 1 0 x < 1 f(x) = 0 2 x < 0, 1 x < 2 x = 1 a 0 = 1 2 a n = f(t) dt = f(t) cos nπt 2 dt = dt = 1 2, 1 = ( 1)k, (n = 2k + 1). (2k + 1)π cos nπt 2 b n = 1 f(t) sin nπt dt = 1 sin nπt n = 4k, 1 (4k+1)π n = 4k + 1, = 1 (2k+1)π n = 4k + 2, 1 (4k+3)π n = 4k + 3. S[f] = a a n cos nπx 2 + b n sin nπx 2. n=0 dt = sin nπ/2 nπ dt = 1 cos nπ/2 nπ

70 64 2 Fourier 2.24 Haar scaling [ 1, 1) f(x) = 1 0 x < x < 0 Fourier [ π, π) f(x) = 1 0 x < π 1 π x < 0 (2.36) 2π (square wave) 2.10 Fourier Ç å π b n = 1 f(t) sin nt dt = 1 π π π 0 n = 2k = 4 n = 2k + 1. (2k + 1)π S[f] = k=0 0 π + π 0 sin nt dt = 2 π π 0 sin nt dt = 2 π 4 sin(2k + 1)x (2.37) (2k + 1)π 11 S N=11 [f] = 4 sin x π + 4 sin 3x 3π + 4 sin 5x 5π + 4 sin 7x 7π + 4 sin 9x 9π + 4 sin 11x 11π..., π, 0, π,... Fourier lim ϵ 0 2 (f(0 ϵ) + f(0 + ϵ)) = 1 (f(0+) + f(0 )) 2 1 cos πn n

71 2.7 Fourier (a) Fourier N S N (x) N = 3, 5, 7, 9, 11 N N f(x) overshoot 2.12(b) Gibbs (Gibbs phenomena) (a) (b) 2.12 (a) [ π, π) Fourier N = 3, 5, 7, 9, 11 S N (x) [ 4, 4] (b) S 61(x) π, 0, π N -1 1 N overshoot Gibbs 2.25 Mathematica [ π, π) Fourier S N (x) N overshot f[x_] := Piecewise[{{-1, -Pi <= x && x < 0}, {1, 0 <= x && x < Pi}}](*

72 66 2 Fourier *) fs[x_, n_] := FourierSinSeries[f[x], x, n] (* n Fourier Sine *) f3[x_] = fs[x, 3]; f5[x_] = fs[x, 5]; f7[x_] = fs[x, 7]; f9[x_] = fs[x, 9]; f11[x_] = fs[x, 11]; Plot[{f[x], f3[x], f5[x], f7[x], f9[x], f11[x]}, {x, -4, 4}] (* *) [0, T ) 1 0 x < T/2 f T (x) = 0 T/2 x < T Fourir a 0 = 1, a n = 0, (n 0), 0, n = 2k b n = 2 πn, n = 2k + 1. Fourier S[f T ] = π k=0 1 2k + 1 2π(2k + 1) sin x T 2.26 f T (x) Fourier a Fourier (2.26) (2.27) a = T/ [ π, π) f(x) = x, π x < π (2.38)

73 2.7 Fourier 67 {±(2k + 1)π} k=0,1,... (sawtooth wave) Fourier b n = 2 π π 0 = 2( 1)n+1 n x sin nt dt = 2 cos nπ n Fourier 2.13 (2.39) 2( 1) n+1 S[f] = sin nx n n= [ π, π) Fourier 60 S N=60[f] [ 4, 4] 2.12 π, π overshoot Gibbs 2.27 Mathematica [ π, π) Fourier S N=60 [f] 2.13

74 68 2 Fourier 2.8 Gibbs * 1 2π [ π, π) f(x) = π x 0 x < π π x π x < 0 (2.40) 2.14 Gibbs 2.4 Dirichlet [ π, π) x = 0 {2πn} n=0,±1,±2,... 2π f Fourier b k b k = 2 π = 2 k π 0 f(x) sin kx, dx = 2 π π 0 (π x) sin kx dx f Fourier S[f] = 2 k=1 sin kx k *1

75 2.8 Gibbs 69 Dirichlet (2.33) N Fourier S N (x) S N(x) = 2πD N (x) 1 S N (x) = x 0 (2πD N (t) 1) dt = x 0 sin ( N + 2) 1 t sin t dt x 2 [0, π) S N (x) f(x) g N (x) g N (x) = S N (x) (π x) = 2π x 0 D N (t) dt π g N (x) = 2πD N(x) g N (x) { µ N g N (µ N ) = 0 } x = 0 µ 1 N = π N lim N N µ 1 N = π N f(x) overshoot g N (x) x 0+ g N (µ 1 N ) 2.28 x 0 sin ( N + 2) 1 t x sin ( N + 1 Å ã sin t dt = 2 2) t 1 dt + O, N. 2 0 t N 2.22 Riemann-Lebesgue d(x) = 1 sin x 2 2 x d(0) = 0 [0, π) d(x) C 1 x Å d(x) sin N + 1 ã Å ã 1 t dt = O, N 2 N 0

76 70 2 Fourier x < π Bernoulli B n *2 x sin x = 1 + n=1 (2 2n 2)B n x 2n, (2n)! ( (2 2n 2)B x n 2 d(x) = (2n)! n=1 ) 2n 1 = 1 6 x ( x ) 3 31 ( x ) d(0) = 0 x sin ( N + 1 Å ã S N (x) = 2 2) t 1 dt x + O, N. 0 t N 2.29 N S N (x) (0, π) f(x) lim S N(x) = f(x), 0 < x < π. N lim S N(x) = lim N = 2 x 2 N 0 0 sin t t sin ( N + 1 2) t t dt x = 2 Si( ) x N + 1 (N+ 1 2 )x 2 sin t dt x = lim dt x N N 0 t Si(x) Si( ) = π 2 *2 Bernoulli B n B n = 2(2n)! (2π) 2n r=1 1 r 2n = (2n)! (2 2n 1 1)π 2n r=1 ( 1) r 1 2(2n)! r 2n = (2 2n 1)π 2n r=0 1 (2r + 1) 2n. B n p 1 2n p B 1 = 1/6, B 2 = 1/30, B 3 = 1/42, B 4 = 1/30, B 5 = 5/66, B 6 = 691/2730, B 7 = 7/6, B 8 = 3617/510, B 9 = 43867/798

77 2.8 Gibbs 71 x 0+ g N (x) overshoot g N (x N ) g N (x) x sin ( N + 1 Å ã g N (x) = S N (x) (π x) = 2 2) t 1 dt π + O, N. 0 t N 2.28 Å 1 g N (x) = 2 sup Si(t) π + O N sup 0 x<π 0 t< 2.22 (iv) lim sup N 0 x<π g N (x) = 2 Si(π) π = ã, N. N N x N = π x N 0+ overshoot g N (x N ) 2.15 x sin ( N + lim g N (x) = 2 2) 1 t dt π N N x N =π 0 t π sin t = 2 dt π 0 t = 2 Si(π) π = x = 0 g N (x) µ 1 N lim N N µ 1 N = π f(x) overshoot

78 72 2 Fourier 2.15 Fourier S N=60[f] N S N x = 0 Gibbs x = 0 overshoot 2.30 ( Gibbs ) (2.40) [ π, π) f Fourier N S N (x) f(x) overshoot g N (x) = S N (x) f(x) (i) (0, π) lim N S N (x) = f(x). (ii) S N N overshoot overshoot N (iii) lim N g N (x) = 2 Si(π) π, 0 x < π. (iv) x 0+ overshoot lim N g N (x N ) = N x N =π 2 Si(π) π = Mathematica 2.15 Gibbs

79 2.9 Fourier Fourier Fourier L Gibbs Fourier Riemann Lebesgue f Fourier Fourier 2.32 (Riemann Lebesgue ) I = [a, b] f lim k b a f(x) cos kx dx = lim k b a f(x) sin kx dx = 0. (2.41) I I k m k = inf x Ik f(x), M k = sup x Ik f(x) I s S m m s = m i I k, S = M i I i. i=1 i=1 f f(x) M(x I). ϵ > 0 δ > 0 < δ 0 S s < ϵ 2 : a = x 0 < x 1 < < x m = b b f(x) cos kx dx a = m xi (f(x) f(x i ) cos kx dx i=1 x i 1 + m xi f(x k ) cos kx dx i=1 x i 1 m m 1 (M i m i )(x i x i 1 ) + M k sin kx i sin kx i 1 i=1 (S s ) + 2mM k i=1 < ϵ 2 + 2mM k

80 74 2 Fourier 1 m k 0 = 4mM ϵ k k 0 b f(x) cos kx dx < ϵ 2 + ϵ 2 = ϵ a k cos / sin kx f x I f k b a ï sin kx f(x) cos kx dx = k Ç = 1 k ò b b f(x) a a sin kx f (x) dx k (f(b) sin kb f(a) sin ka) b a f (x) sin kx dx å Fourier 2π f Fourier S N (x) S N (x) = a N a k cos kx + b k sin kx k=1 Fouerier (2.1) (2.2) a k = 1 π b k = 1 π π π π π f(x) cos kx dx, k 0, f(x) sin kx ( Fourier ) 2π f x Fourier f(x) f(x) = a lim k (a k cos kx + b k sin kx)

81 2.9 Fourier 75 N S N (x) f(x) Foueir S N (x) = 1 π f(t) dt + 1 2π π π ( π 1 f(t) = 1 π = 1 π = π π π π π f(t) ( 1 N Å π k=1 π π f(t) cos kt dt cos kx + ) N 2 + cos kt cos kx + sin kt sin kx k=1 ) N 2 + cos k(t x) dt k=1 f(t)d N (t x) dt D N (x) (2.32) Dirichlet D N (x) = Å sin N + 1 ã x 2 2π sin x 2 π ã sin kt dt sin kx u = t x S N (x) S N (x) = π π (2.34) π π D N (t) dt = 1 S N (x) f(x) = g(t) = f(t + x)d N (t) dt. (2.42) π = 1 2π π π f(x + t) f(x) sin t 2 (f(x + t) f(x)) D N (t) dt π = f(x + t) f(x) sin t 2 f(x + t) f(x) t 2 Å sin N + 1 ã t dt 2 t 2 sin t 2

82 76 2 Fourier [ π, π] t = 0 t = 0 g(0) = 2f (x) lim t 0 t/ sin t = 1 Riemann Lebesgue 2.32 S N (x) f(x) = π (f(x + t) f(x)) D N (t) dt 0, N. π Fourier 2.34 x f f(x + 0) f(x 0) f(x + 0) = lim f(x + h) h>0 h 0 f(x 0) = lim f(x h) h>0 h 0 f x f +(x) f(x + h) f(x) = lim h>0 h h 0 f (x) = lim h<0 h 0 f(x + h) f(x) h x f(x + 0) = f(x ) [ π, π) x f(x) x =,..., ±π, ±2π,... f(π 0) = π, f (π) = 1 f(π + 0) = π, f +(π) = [0, π) x = π 2 f ( π 2 ) = 1 f +( π 2 ) = ( Fourier ) 2π x f x Fourier F (x) F (x) = f(x + 0) + f(x 0) 2

83 2.9 Fourier 77 x 2.33 π Dirichlet D N (x) D N (u) du = 0 π D N (u) du = 1 2 S n (x) (2.42) N π π f(t + x)d N (t) dt f(x + 0) + f(x 0) 2 (2.43) 2 π 0 0 π f(t + x)d N (t) dt f(t + x)d N (t) dt f(x + 0), 2 f(x 0) N π 0 0 π N (2.43) (f(t + x) f(x + 0))D N (t) dt 0, (2.44) (f(t + x) f(x 0))D N (t) dt 0 (2.45) (2.44) Dirichlet 1 π Ç å Å f(x + u) f(x + 0) 2π 0 sin u sin N + 1 ã u du Riemann Lebesgue 2.32 u u 0+ f 0 (2.45) [ π, π) f(x) = x f f(π + 0) = π, f(π 0) = π S N (±π) = 0

84 78 2 Fourier Fourier Fourier 1.63 {F n } F [a, b] (uniform convergence) ε > 0 x [a, b] N = N(ε) n N F n (x) F (x) < ε N ε x f(x) Fourier f(x) S N (x) = a a k cos kx + b k sin kx k=1 N f(x) Gibbs 2.38 ( ) piecewise smooth π ±nπ, n Z 2.40 ±nπ/2, n Z [ a, a] 2a f(x) Fourier f(x) f 2 [ π, π]

85 2.9 Fourier 79 f f Fourier f(x) = n f (x) = n a n cos nx + b n sin nx, a n cos nx + b n sin nx Fourier a n = a n n 2, b n = b n n 2 a n = 1 π π = f(x) π f(x) cos nx dx sin nx n π 1 π π π π sin nx f(x) dx n = b n n b n f Fourier = f cos nx (x) n 2 π 1 π π π n 2 f (x) cos nx dx π = a n n 2 π (2.46) (2.47) f Riemann-Lebesgue 2.32 a n, b n n 0 a n, b n M a n + b n = n=1 n=1 a n + b n n 2 n=1 M + M n 2 = 2M n=1 1 n 2 < + dx/x 2 1 n=1 1/n f(x) Fourier F (x) = a a k cos kx + b k sin kx k=1

86 80 2 Fourier a k + b k < + k=1 Fourier f(x) a k cos kx + b k sin kx a k + b k f Fourier x k a k + b k F (x) S N (x) = a N a k cos kx + b k sin kx F (x) S N (x) = x k=1 F (x) S N (x) k=n+1 k=n+1 a k cos kx + b k sin kx a k + b k N ε > 0 N 0 N > N 0 F (x) S N (x) < ε, N > N 0. N 0 x k=1 a k cos kx + b k sin kx S N (x) 2.40

87 2.9 Fourier Fourier L 2 f Fourier Fourier L f L 2 ([ π, π]) V N = span{1, cos kx, sin kx} 1 k N f Fourier a k, b k f N (x) = a N a k cos kx + b k sin kx k=1 f N V N L 2 f f f N L 2 = min g V N f g. f N V N 1.42 f N f 2.43 L 2 ([ π, π]) 2π L 2 ([ π, π]) f 2.16(a) f x = c (c ϵ, c + ϵ) g(x) 2.16(b) f g (c ϵ, c + ϵ) (c ϵ, c + ϵ) g L 2 ([ π, π]) f

88 82 2 Fourier (a) (b) 2.16 (a) [ π, π) f(x) (b) g(x) f g 2.44 f L 2 ([ π, π]) f Fourier a k, b k f N (x) = a N a k cos kx + b k sin kx k=1 Fourier α k = 1 2π π π f(x)e ikx dt, k Z

89 2.9 Fourier 83 N f N (x) = α k e ikx k= N f N L 2 ([ π, π]) f f f N L 2 0, N. f L 2 ([ π, π]) 2.43 ϵ > 0 g f g L 2 < ϵ (2.48) g Fourier c k, d k V n g N (x) = c N c k cos kx + d k sin kx k= N 0 x [ π, π] N > N 0 g(x) g N (x) < ϵ g g N 2 = π π π π g(x) g N (x) 2 dx ϵ 2 dx = 2πϵ, N > N 0, (2.48) f g N f g + g g N < ϵ + 2πϵ, N > N f N f V N f f N f g N (1 + 2π)ϵ, N > N 0. Parseval Fourier 1.60

90 84 2 Fourier 2.45 (Parseval ) [ π, π] f Fourier a a k cos kx + b k sin kx k=1 L 2 ([ π, π]) 1 π π π f(x) 2 dx = a a k 2 + b k 2. (2.49) k=1 [ π, π] f Fourier k= α k e ikx L 2 ([ π, π]) 1 2π f(x) 2 = 1 π f(x) 2 dx = 2π π f, g L 2 ([ π, π]) 1 1 π f g = f (x)g(x) dx = 2π 2π π k= k= α k 2. (2.50) α kβ k. (2.51) (2.51) f, g Fourier N N f N (x) = α k e ikx, g N (x) = β k e ikx k= N k= N 2.44 N L 2 ([ π, π]) f N f, g N g N f N g N = α k e ikx k= N N = 2π a kβ k k= N N k= N N N β k e ikx = αkβ l e ikx e ilx k= N l= N

91 2.9 Fourier 85 (2.51) f N g N f g, N f g f N g N = ( f g f g N ) + ( f g N f N g N ) f g g n + f f N g N f g g N + f f N g N Schwartz L 2 ([ π, π]) f N f 0, g g N 0 N { e n } n=1,...,n V N f { e n f } Bessel (1.33) (2.50) N α k 2 f 2. k= N [ π, π) f(x) = x Fourier (2.39) b n = 2( 1)n+1. n 1 π x 2 dx = 2 π π 3 π2, b n 2 4 = n 2 n=1 n=1 (2.49) Riemann zeta ζ(s) s = 2 Euler Basel 1735 π 2 6 = 1 n 2 n=1

92

93 87 3 Fourier 3.1 Fourier 3.1 R L 1 Lebesgure f(x) dx < + (3.1) 3.2 (Fourier ) R f(x) Fourier Fourier transformation f(λ) = 1 2π f(x)e iλx dx (3.2) 3.3 f(x)e iλx = f(x) (3.2) f(λ) < λ < f(λ) 1 2π f(x) dx 1, x f(x) = 0, x > 1 Fourier

94 88 3 Fourier Si(x) 2.22 Si( ) = π 2 = f(λ) = 1 2π f(x)e iλx dx = 1 2π 1 1 e iλx dx 1 = iλ [ e iλx ] 1 1 2π = 1 iλ 2π (e iλ e iλ ) 2 sin λ = π λ. e kx, x f(x) = Fourier 0, x < 0 f(λ) = 1 e kx e iλx dx = 1 e (k+iλ)x dx 2π 2π 0 1 î = ó e (k+iλ)x 2π(k + iλ) = 1 2π(k + iλ) f(x) Fourier f(λ) f(x) f(λ) f(λ) f(λ) l λ 1 2π x l f(x)e iλx dx = 1 l 2π f(λ + λ) f(λ) = 1 2π < 2ϵ + 1 2π l f(x)e iλx dx + 1 f(x)e iλx dx 2π < ϵ f(x)e i(λ+ λ)x dx 1 2π l l f(x)e iλx dx f(x) î e i(λ+ λ)x e iλxó dx.

95 3.1 Fourier 89 e i(λ+ λ)x e iλx = ix λe i(λ+θδλ)x, 0 < θ < 1 f(λ + λ) f(λ) < 2ϵ + 1 l λ l f(x) dx 2π l λ < Ç 1 l 1 l f(x) dxå ϵ 2π l f(λ + λ) f(λ) < 3ϵ f(λ) Fourier Fourier Fourier [ l, l] f Fourier l 2.9 Fourier f(x) f(x) = α n e inπx/l (3.3) n= α n = 1 2l l f(t)e nπx/l l f l Fourier

96 90 3 Fourier λ n = nπ l δλ = λ n+1 λ n = π l f(x) = lim l [ n= [ = lim l [ = lim l n= n= F l (λ) = 1 l f(t)e iλn(x t) dt 2π l (3.4) [...] n= F l (λ n )δλ Ç 1 l å ] f(t)e inπt/l dt e inπx/l 2l l ] 1 l f(t)e inπ(x t)/l dt 2l l ] 1 l f(t)e iλn(x t) dt δλ (3.4) 2π l F l(λ)dλ Riemann (3.4) f(x) = lim F l (λ)dλ l F l (λ) l 1 2π f(x) = 1 2π = 1 2π f(t)eiλ(x t) dt f(t)e iλ(x t) dtdλ Å 1 2π ã f(t)e iλt dt e iλx dλ (3.5) f(λ) = 1 2π f(t)e iλt dt f Fourier (3.5)

97 3.1 Fourier (Fourier ) f(x) (, ) f n4 f (n) (x) lim x ± xm f (n) (x) = 0 (m, n = 0, 1, 2,... ) f(a) = 1 Å ã dλ e iλa dx e iλx f(x) 2π (3.5 ) Fourier 3.7 Fourier (3.5) Fourier 3.2 f(x) = 1 2π f(λ)e iλx dλ 3.8 f(x) R f(x) Fourier f(λ) lim l 1 l 2π l f(λ)e iλx λ = f(x 0) + f(x + 0) 2 (3.6) f(x) R (3.6) a x b f(x) Fourier f(λ) = 1 2π f(x)e iλx dx f(x) = 1 2π f(λ)e iλx dλ (3.7) f(x) fλ) f(x) Fourier (3.3) f(x) Fourier lim N N n= N α n e inπx/l = f(x 0) + f(x + 0) 2

98 92 3 Fourier Fourier f(x) Fourier f(λ) (3.7) f(λ) = 1 f(x)e iλx dx f(x) = f(λ)e ±iλx dλ (3.7 ) 2π f(ξ) = f(x)e 2πiξx dx f(x) = f(ξ)e 2πξx dξ (3.7 ) f f f(x) f(ξ) f(λ)) f(x) Fourier f(λ) f cos 3x, x π 3.11 f(x) = χ [ π,π] (x) cos 3x = Fourier 0, x > π χ I I f f(λ) = 1 2π = f(x) cos λx dx = 1 2π π π Å ã 2λ sin λπ 1 π(9 λ2 ) O, (λ ). λ cos 3x cos λx dx f(λ) λ = ±3 3.1(a) f 3 sin 3x, x π 3.12 f(x) = χ [ π,π] (x) sin 3x = Fourier 0, x > π f(λ) 3.1(b) f R f(λ) O ( 1 λ 2 ), (λ )

99 3.1 Fourier 93 (a) 3.1 (b) (a) f(x) = χ [ π,π] (x) cos 3x χ I I Fourier f(λ) x = ±π R f(λ) λ = ±3 3 f(λ) O ( 1 λ), (λ ) (b) f(x) = χ[ π,π] (x) sin 3x Fourier f(λ) f(λ) O ( 1 λ 2 ), (λ )

100 94 3 Fourier 1, x π 3.13 f(x) = χ [ π,π] (x) = Fourier f(λ) 0, x > π x = ±π R f f(λ) = 1 1 2π = 2 sin λπ O π λ π f(x)e iλx dx = 1 1 2π Å ã 1, (λ ). λ π cos λx dx f(λ) λ f(x) = χ [ π,π] (x) Fourier 1 2π 2 sin λπ πλ e iλx dλ = 1, x < π, 1 2, x = ±π, 0, x > π x = ±π 3.8 Fourier 2 sin λx πλ dλ = + x + π, x π, 3.15 f(x) = (1 x )χ [ π,π] (x) = π x, 0 < x π, 0, x > π Fourier f(λ) 3.3 f f(λ) = 2 2π = f(x) cos λx dx = 2 π (π x) cos λx dx 2π Å ã 2 1 cos λπ 1 π λ 2 O λ 2, (λ ). 0

101 3.1 Fourier f(x) = (1 x )χ [ 1,1] (x) Fourier f(λ) f(λ) O ( ) 1 λ, (λ ) f(x) = χ [ π,π] sin 3x(x), f(x) = (1 x )χ [ π,π] (x) R f(λ) λ O ( 1 λ 2 ) f(x) = χ [ π,π] (x) cos 3x, f(x) = χ[ π, π](x) x = ±π Fourier f(λ) λ O ( 1 λ) Fourier a n, b n O ( 1 n 2 ) 2.7 Fourier a n, b n O ( 1 n) 3.17 f(x) = Fourier 0, x < 0 e ax, x 0, Re(a) > 0 f(λ) = 1 2π 1 a + iλ

102 96 3 Fourier 3.2 Fourier f(x) Fourier f(λ) f F f(λ) = 1 2π f(x)e iλx dx =F[f](λ) (3.2 ) Fourier F 1 F 1 [g](x) = 1 2π g(λ)e iλx dλ (3.8) 3.8 Fourier F 1 F F 1[ F[f] ] = f (3.7 ) F 1[ F[f] ] = F 1 [ f](x) = 1 2π F f(λ)e iλx dλ F 1 = f(x) 3.8 Fourier. Fourier 3.18 [Fourier ] f g R x f(x) = 0 (1) Fourier Fourier c F[f + g] = F[f] + F[g], F[cf] = cf[f] F 1 [f + g] = F 1 [f] + F 1 [g], F 1 [cf] = cf 1 [f]. (2) f x n Fourier F[x n f(x)](λ) = i n dn F[f](λ) (3.9) dλn

103 3.2 Fourier 97 (3) f λ n Fourier F 1 [λ n f(λ)](x) = ( i) n dn dx n F 1 [f](x) (3.10) (4) f n Fourier F[f (n) (x)](λ) = (iλ) n F[f](λ). (3.11) (5) f n Fourier F 1 [f (n) (λ)](x) = ( iλ) n F 1 [f](x). (3.12) (6) f x a Fourier F[f(x a)](λ) = e iλa F[f](λ). (3.13) (7) f bx Fourier F[f(bx)](λ) = 1 b F[f] Å λ b ã (3.14) (8) x < 0 f(x) = 0 F[f](λ) = 1 2π L[f](iλ), (3.15) L[f] f Laplace L[f](s) = 0 f(x)e xs dx 1. Fourier F[f + g](λ) = 1 [f(x) + g(x)]e iλx dx 2π = 1 2π =F[f](λ) + F[g](λ) f(x)e iλx dx + 1 2π g(x)e iλx dx

104 98 3 Fourier F[cf]cF[f] Fourier f x n Fourier F[x n f(x)](λ) = 1 2π x n f(x)e iλx = (i) n dn dλ n f(x)e iλx x n f(x)e iλx dx ß F[x n f(x)](λ) = (i) n dn 1 dλ n f(x)e iλx dx 2π =(i) n dn dλ n F[f](λ). Fourier f n Fourier F[f (n) (x)](λ) = 1 2π f (n) (x)e iλx dx f 1 2π f (n) (x)e iλx dx = 1 2π f (n 1) (x) d dx e iλx dx 1 = (iλ) f (n 1) (x)e iλx dx 2π n 1 1 2π f (n) (x)e iλx dx = (iλ) n 1 2π = (iλ) n F[f](λ). f(x)e iλx dx F[f(bx a)](λ) = 1 b e iλa/b F[f](λ/b). (3.16)

105 3.2 Fourier 99 Fourier s = bx a dx = ds/b F[f(bx a)](λ) = 1 2π = 1 2π F[f(bx a)](λ) = e iλa b = e iλa b f(bx a)e iλx dx f(s)e 1 2π 1 b F[f] Å λ b s+a iλ( b ) fracdsb. f(s)e iλ b s ds b ã 8.Laplace 3.19 f(x) = χ [ π,π] (x) sin 3x Fourier 3.12 f(λ) = 3 2i sin λπ π(9 λ2 ) [ π, π] f = 3 cos 3x (1) 3.11 Fourier f (λ) = 3λ 2i sin λπ π(9 λ2 ) (3.17) f (λ) 3.18(4) f(λ) f (λ) = iλ f(λ) = iλ 3 2i sin λπ π(9 λ2 ) (3.17)

106 100 3 Fourier (a) (b) 3.4 (a) f 2(x) = χ [ π/2,π/2] (x) sin 6x f(x) = χ [ π,π] (x) sin 3x 3.12 f 2(x) = f(2x) 3.1(a) x 1/2 (b) Fourier F[f 2(x)](λ) = F[f(2x)](λ) 3.1(b) λ = 3 f(λ) λ = 6 1/2 F[f(2x)](λ) f(λ) λ 2 1/2 (1/2) f(λ/2) 3.20 Fourier F[f(bx)](λ) = 1 Å ã λ b F[f] b (3.14 ) f Fourier

107 3.2 Fourier 101 b > 1 f(bx) f(x) x 1/b 3.12 f(x) = χ [ π,π] (x) sin 3x f(2x) = χ [ π/2,π/2] (x) sin 6x x 1/2 3.4(a) Fourier F[f(2x)](λ) 3.4(b) F (λ) λ = 3 λ = 6 F[f(2x)](λ) = 1/2 f(λ/2) f x Fourier x f Fourier V W T : V W T : W V T w v V = w T v W Fourier 3.18 Fourier F Fourier F Fourier Fourier F (Fourier ) f g L 2 2 F[f] g L 2 = f F 1 [g] L 2. (3.18) F[f] g L 2 = = 1 2π = = f (λ)g(λ) dλ f (t) f (t)e iλt dt g(λ) dλ Å 1 2π f (t)f 1 [g](t) dt = f F 1 [g] L 2. ã g(λ)e iλt dλ dt F 1 f

108 102 3 Fourier Plancherel Fourier L (Plancherel ) f g L 2 F[f] F[g] L 2 = f g L 2 F 1 [f] F 1 [g] L 2 = f g L 2 F[f] L 2 = f L 2. F[f] F[g] L 2 = f F 1 F[g] L = f g L [ π, π] Fourier Parseval π π π f(x) 2 dx = a π f(x) 2 = 1 2π π π a k 2 + b k 2 (2.49 ) k=1 f(x) 2 dx = k= α k 2 (2.50 ) Parseval ( f g ) f(x) g(x) convolution f g(x) f g(x) = f(t)g(x t) dt (3.19)

109 g(x) (1/a)χ [ a/2,a/2] (x) a > 0 f g(x 0 ) x 0 a f(x) f(x) f(x) g g(x) = (1/a)(1 x /a)χ a,a] (x) f g(x 0 ) x 0 f(x) x 0 x 0 f g(x) g f(x) 3.24 (3.19) y = x t f g(x) = f(x t)g(t) dt f g(x) = g f(x) f g(x) f(x) g(x) 3.25 ( ) f g L 1 Fourier F[f g] = 2π f ĝ, (3.20) F 1 [ f ĝ] = 1 f g. 2π (3.21) ) Fourier F[f g](λ) = 1 (f g)(x)e iλx dx 2π = 1 2π f(x t)g(t) dt e iλx dx. e iλx = e iλ(x t) e iλt F[f g](λ) = 1 2π = 1 2π = 2π = 2π f ĝ Å 1 2π f(x t)e iλ(x t) g(t) dt e iλt dt f(s)e iλs g(t) ds e iλt dt ã Å 1 ã f(s)e iλs ds g(t)e iλt dt 2π

110 104 3 Fourier ) 3.8 2πF 1 [ f ĝ] = F 1 [F(f g)] = f g 3.26 Mathematica f(x) g(x) f g(x) Convolve Convolve[f[y], g[y], y, x] y x 1.29 Haar ϕ(x) Gauss e x2 haar[x_] := If[0 <= x <= 1, 1, 0] conv[x_] = Convolve[haar[y], Exp[-y^2], y, x] 1/2 Sqrt[\[Pi]] (Erf[1 - y] + Erf[y]) erf[y] erf(y) = 2 y π 0 e t2 dt 3.4 *1 Fourier signal f : R C filter f f L f, g a, b L[af + bg] = a L[f] + b L[g] f f a f a (x) = *1

111 f(x a) f x t f(x) f(t) x R 3.27 shift-invariant R f a L[f a ] = (Lf) a, x L[f a ](x) = (Lf)(x a) f a f a 3.28 l(x) x l(x) = 0 f (Lf)(x) = (l f)(x) = L l(x y)f(y) dy (Lf)(x a) = = = = L[f a ](x) l(x a y)f(y) dy l(x y )f(y a) dy l(x y )f a (y ) dy y = y + a L 3.29 l(x) l(x) = 1 0 x 0 x < 0

112 106 3 Fourier L ( Lf)(x) = L[f a ](x) = = l(x y)f(y) dy = x 0 x 0 x a a ( Lf)(x a) = f(y a) dy f(y )dy y = y a. x a 0 f(y) dy f(y) dy. L[f a ](x) ( Lf)x a) λ e λx 3.30 L λ h ĥ L(e iλx ) = 2π ĥ(λ)eiλx. h(x) system function transfer function h λ (x) = L(e iλx ) L a L[e iλ(x a) ] = h λ (x a). L L[e iλ(x a) ] = h λ (x a) = L[e iλa e iλx ] = e iλa L[e iλx ] = e iλa h λ (x).

113 h λ (x a) = e iλa h λ (x) (3.22) a = x h λ (0) = e iλx h λ (x) h λ (x) = h λ (0)e iλx ĥ(λ) = hλ (0)/ 2π (3.22) Fourier 3.18 (6) F[f(x a)](λ) = e iλa F[f](λ) (3.13 ) λ (3.22) h λ (x) h(x) Fourier F[f](λ) = ĥ(λ) 3.31 L L(f) = f h h f Fourier 3.8 f(x) = F 1 [ f] = 1 2π L Å 1 (Lf)(x) = L 2π f(λ)e iλx dλ. ã f(λ)e iλx dλ L Riemann OK (Lf)(x) = 1 2π = 1 2π f(λ)l[e iλx ](x) dλ f(λ) Ä 2π ĥ(λ)ä e iλx dλ 3.30 = 2πF 1 î f(λ) ĥ(λ) ó (x) 3.8 = (f h)(x) 3.25

114 108 3 Fourier 3.32 f δ(x) L (Lf)(x) = δ h(x) = = h(x) δ(x y)h(y) dy h(x) L h(x) inpulse response function (3.22) λ e iλx L L[e iλx ] = 2πĥ(λ)eiλx Lf = f h L[f](λ) = 2π f ĥ(λ) (3.23) f λ c λ > λ c h λc (x) Fourier 1 λ λ c, ĥ λc (λ) = 2π (3.24) 0 λ > λ c

115 Fourier hλc (λ) (3.24) x 0 f xc (x) = χ [0,xc](x) (Lf xc )(x) x < 0 0 x ĥλ c (λ) Fourier h λc (λ) = F 1 [ĥλ c ] = 1 2π = 1 2π λc iλx dλ ĥ λc (λ)e λ c e iλx dλ = eiλcx e iλcx 2iπx = sin λ cx πx f xc f xc (x) = χ [0,xc](x) = 1 0 x xc, 0 x < 0 x > x c

116 110 3 Fourier L Lf xc 3.31 (Lf xc (x) = (f xc h c )(x) = = xc = 1 π 0 λcx f xc (y)h c (x y) dy sin λ c (x y) π(x y) λ c(x x c) sin w w dw = 1 π ( Si(λc x) Si(λ c (x x c )) ). si(x) x f xc x < 0 x 3.33 causal x < 0 f(x) = 0 Lf x < 0 (Lf)(x) L h Lf = f h L x < 0 h(x) = 0 L h(x) Fourier ĥ(λ) ĥ(λ) = L[h](iλ) 2π L[g](s) g Laplace L[g](s) = 0 g(x)e xs dx x < 0 h(x) = 0 x < 0 f(x) = 0 x < 0 (Lf)(x) = 0 (Lf)(x) = (f h)(x) = 0 f(y)h(x y) dy

117 x < 0 y 0 h(x y) = 0 x < 0 (Lf)(x) = 0 h(x) Fourier ĥ(λ) = 1 2π h(x)e iλx dx h x < 0 h(x) = 0 ĥ(λ) = 1 h(x)e iλx dx 2π 0 = L[h(x)/ 2π](iλ) (a) (b) 3.6 (a) f(x) = e x/3( sin 3x + 2 sin 5x sin 2x sin 40x ) [0, π] (b) f h(x) [0, π] (a) 3.35 (Butterworth ) [22, 6 2 F ] S.Butterworth 1930 low pass a, b h h(x) = a e bx x 0 0 x < ĥ(λ) = 1 (Lh)(iλ) = 1 a 2π 2π b + iλ

118 112 3 Fourier (3.23) L[f](λ) = 2π f ĥ(λ) ĥ(λ) λ hight cut low pass λ ĥ(λ) ĥ(0) = b b2 + λ 2 f 3.6(a) f(x) = e x/3( sin 3x + 2 sin 5x sin 2x sin 40x ) (3.25) a = b = 10 f h 3.6(b) f h(x) = e x/3( cos 3x cos 5x cos 38x cos 42x sin 3x sin 5x sin 38x sin 42x ) 3.6(b) (a) (b) Mathematica ( ) f(x) L 2 [R] band limitted Ω > 0 Fourier f [ Ω, Ω] λ > Ω f(λ) = 0 2Ω Ω > 0 µ = Ω 2π Nyquist Nyquist frequency 2µ = Ω/π Nyquist 3.38 (Shannon-Whittaker ) f f(λ) 2Ω λ > Ω f(λ) = 0

119 f = F 1 [ f] π/ω x k = πk/ω, k = 0, ±1 ± 2,... {f(x k )} f(x) = k= f Å ã kπ sin(ωx kπ) Ω Ωx kπ (3.26) Fourier Fourier a = Ω, x = λ f(λ) [ Ω, Ω] Fourier f(λ) = l= c l e iπlλ/ω, c l = 1 Ω f(λ)e iπlλ/ω dλ 2Ω Ω λ Ω f(λ) = 0 c l (, ) c l = 2π 2Ω 1 2π f(λ)e iπlλ/ω dλ. 3.8 Fourier c l c l = 2π 2Ω f( lπ/ω ) c k Fopurier l k = l 2π f(λ) = 2Ω f( kπ/ω ) e iπkλ/ω (3.27) k= f 2.40 (3.27) 3.8 Fourier f(x) = 1 2π = 1 2π Ω Ω f(λ)e iλx dλ f(λ)e iλx dλ f(λ)

120 114 3 Fourier (3.27) f f(x) = k= 2π 2Ω f( kπ/ω ) 1 2π Ω Ω (3.28) πkλ i e Ω +iλx dλ (3.28) Ω Ω e πkλ i Ω +iλx sin(ωx kπ) dλ = 2Ω Ωx kπ (3.26) 3.39 (3.26) O ( j 1) over sampling O ( j 2) Nyquist f (3.26) aliasing f(x) 3.7(a) f(x) = e 0.2t2 cos 3t f(x) Fourier f(λ) Mathematica packet[t_] := Exp[-0.2 t^2] Cos[3 t] ftpacket[lambda_] = FourierTransform[packet[x], x, lambda] f(λ) = 1 2π f(x)e iλx dλ = ( cosh 1.25(λ 3) 2 sinh 1.25(λ 3) 2) ( cosh 1.25(λ + 3) 2 sinh 1.25(λ + 3) 2)

121 f(λ) 3.7(b) λ > 6 f(λ) 0 f(λ) = 0 λ ± (a) (b) 3.7 (a) f(x) = e 0.2t2 cos 3t (b) f(x) Fourier f(λ) λ > 6 f(λ) 0 λ = ± f(λ) = 0 λ Ω c f(λ) = 0 f(λ) [ Ω c, Ω c ] [ Ω c, Ω c ] 3.38 Ω (3.26) Ω c π/ω c 2n + 1 f (n) Ω c (x) = n k= n Å ã kπ sin Ωc x kπ f Ω c x kπ Ω c (3.29) f(x)

122 116 3 Fourier 3.7(b) f(λ) Ω c = 3 Ω c = 4 (3.29) 3.8 (a) (b) 3.8 (a) Ω c = n = 2 n = 5 f(x) (b) Ω c = n = 2 n = 5 f(x) f (n) Ω c (x) (a) (b) n f(x) 3.8(a) Ω c = f (n) Ω c (x) n = 2 n = 5 f(x) n f(x) 3.8(b) Ω c = f (n) Ω c (x) n = 2 n = 5 f(x) 3.7(b) Ω c = 4 Ω c = 3 n = 2 3.8(a) f(x)

123 Ç Ωc å + f(λ) 2 dλ Ω c Ω c

124

125 M.J. Lighthill [18] * Schwartz ψ suppψ D D χ [7] D χ ϕ D χ ψ (1.9) 4.1 f(x) good function rapidly decreasing function f(x) f (n) (x) x N O ( x N) 1 x N 4.2 f(x) fairly good function temptered distribution f(x) f (n) (x) x m O ( x m) x m *1 M.J. Lighthill [18] 70 L.Schwartz [7] Schwartz

126 120 4 e x2 4.3 generalized function distribution χ(x) h n (x) g(x) lim n h n (x)g(x) dx = χ(x)g(x) dx α(x) β(x) a n (x), b n (x) lim n a n (x)g(x) = lim n b n (x)g(x). f(x) f n (x) = f(x)e x2 /n f n (x) f n (x) e x2 /n lim n f n (x)g(x) dx = f(x)g(x) dx (4.1) D n (x) x = x 0 x = x 0 regular point x = x 0 local value n 2 D n (x) = π e nx (4.1) *2 D n (x) 4.2 δ(x) *2 D n(x) Hermite H m(x) 1.16(3) Rodriguez d m ( 1)m Dn(x) = dxm n m/2 Hm( nx) D n(x)

127 n x 0 D n (x) dx = 1 (4.2) lim D n(x) = 0 (4.3) n x 0 x = 0 D n (0) δ(0) δ(x) x = 0 δ(x) = 0 x (1) α(x) β(x) χ(x) χ(x) = α(x) + β(x). (4.4) α(x), β(x) h n (x) = a n (x) + b n (x) (2) α(x) f(x) χ(x) χ(x) = f(x)α(x) (4.5) α(x) h n (x) f(x) α c χ(x) = c α(x).

128 122 4 [7, p.87] a n (x) b n (x) ( an (x)b n (x) ) f(x) dx (3) χ(x) f(x) dχ(x) dx f(x) = χ(x)f(x) dx. (4.6) α(x) a n (x) h n (x) = d dx a n(x) χ(x) = d dx α(x) h n (x)f(x) dx = = da n (x) f(x) dx dx a n (x) df(x) dx dx. dα(x) dx f(x) dx = (4) χ(x) α(x) df(x) dx. χ(ax + b)f(x) = 1 Å ã x b a χ(x)f. (4.7) a h n (ax + b) x = ax + b

129 [12] Fourier δ(x) = 1 2π dk e ikx (4.8) x 0 δ(x) = 0 dx δ(x) = 1 f(x) f(x) = dx f(x )δ(x x ) 4.4 δ(x) (1) f(x) f(0) = f(a) = δ( x) = δ(x). δ(x)f(x) dx (4.9) δ(x a)f(x) dx (4.9 ) (2) f(x) f(x)δ(x) = f(0)δ(x) (4.10)

130 124 4 xδ(x) = 0 (4.11) (3) Heaviside 1 x > 0 H(x) = 0 x < x > 0 x < 0 θ(x) 1 x > 0 θ(x) = 0 x < 0 dθ(x) dx = δ(x). (4.12) (4) Fourier (5) δ(x x ) = 1 2π 1 = dλ e iλ(x x ) (4.13) δ(x)e iλx dx. (4.14) (4.13) (4.14) δ(x) Fourier 1 δ(ax) = a 1 δ(x) (4.15) (6) F (x) = 0 {x i } δ(f (x)) = i δ(x x i ) df. (4.16) dx x xi δ(x 2 a 2 ) = 1 ( ) δ(x a) + δ(x + a). (4.17) 2a

131 (7) δ (x) = d δ(x) dx xδ (x) = δ(x), δ (x) = δ( x). (1) D n (x) δ(x) lim n D n (x)f(x) dx = δ(x)f(x) dx = f(0) x ξ(x) f(x) = f(0) + f (ξ)x n 2 π e nx f(x) dx f(0) = n 2 df(ξ) π e nx dx dx ß df max n 2 dx π x e nx dx = 1 ß df max n nπ dx 0. (2) 2 D n (x) (1) f(x)δ( x) dx = = f(0) = f( y)δ(y) dy f(x)δ(x) dx. ( f(x)δ(x) ) g(x) dx = f(0)g(0) = ( f(0)δ(x) ) g(x) dx

132 126 4 (3) H(x) x = 0 f(x) dθ(x) dx f(x) dx = = = 0 θ(x) df(x) dx H(x) df(x) dx df(x) dx dx dx dx = f(0). (4) Gauss D n (x) (4.1) n 2 π e nx = 1 e iλx λ2 /4n dλ. 2π e λ2 /4n / 2π 1/ 2π 3.1 Fourier f(x) Fourier f(λ) (3.7) Fourier (3.5) f(x) = 1 2π = dλ e iλx 1 2π dλ f(x ) 1 2π dx e iλx f(x ) dx e iλ(x x ). (4.13), (4.14) δ(x) 1/ 2π Fourier λ = 0 δ(x) dx = 1, (2) f(x) f(x)δ(x) dx = f(0) (5) 4.1.1(4) χ(x) δ(x) f(x) = 1

133 4.3 Fourier 127 (6) ε f(x)δ(x) dx = f(0) ε δ(f (x))g(x) dx = i = i ε δ(f )g(x(f )) df ε df dx g(x i ) df dx x=xi (7) 4.1.1(3) χ(x) δ(x) f(x) = x xδ (x) = δ(x) 1 = δ( x) ( ) = ( x)δ( x) 1 = xδ ( x). 4.3 Fourier ( ) u t = c u x (4.18) t = 0 f(x) u(x, t) = f(x ct) c f(x ct) f(x) lim x ± xm f (n) (x) = 0 (4.18) u(x, t) x Fourier û(λ, t) (4.18) Fourier Fourier (3.11) û (λ, t) = iλû(λ, t)

134 128 4 λ 1 d t) = icλû(λ, t), dtû(λ, û(λ, 0) = f(λ) û(λ, t) = f(λ)e icλt u(x, t) Fourier Fourier (4.13) u(x, t) = F 1 [û](x, t) = 1 2π = 1 2π = dy f(y) = f(x ct) dλ û(λ, t)e iλx dλ f(λ)e iλ(x ct) = 1 2π Å 1 ã dλ e iλ(x ct y) = 2π Å ã dλ e iλ(x ct) dy f(y)e iλy dy f(y)δ(x ct y) 2 u t 2 = c2 2 u x 2 u(x, t) c > 0 u(x, t) (4.19) u(x, 0) = f(x), u t (x, t) = u(x, t)/ t = g(x) (4.19) Fourier 2 u(x, t) d 2 dt 2 û(λ, t) = c2 λ 2 û(λ, t), û(λ, 0) = f(λ), û t (λ, 0) = ĝ(λ) u(x, t) (4.19) u(x, t) = 1 2 (f(x + ct) + f(x ct)) + 1 2c x+ct x ct dy g(y) (4.20)

135 4.3 Fourier (4.20) (4.19) u t = u κ 2 x 2 + α u, κ > 0, α (4.21) x u(x, 0) = f(x) κ (diffusion coefficient) α (drift coefficient) u(x, t) t u(x, t) 0, dx u(x, t) = 1 t = 0 f(x) (4.21) u(x, t) x Fourier û(ξ, t) = u(x, t) = dx e 2πiξx u(x, t) dξ e 2πiξx û(ξ, t) (4.21) Fourier 3.7 u(x, t) < x < 0 t û(ξ, t) 2 d dtû(ξ, t) = ( 4π 2 κξ 2 2πiαξ ) û(ξ, t), û(ξ, t) = f(ξ). (4.22) û(ξ, t) û(ξ, t) = f(ξ) exp [ ( 4π 2 κξ 2 2πiαξ)t ]

136 130 4 u(x, t) u(x, t) = = = = = dξ f(ξ) exp [ ( 4π 2 κξ 2 2πiαξ)t ] exp( 2πiξx) dξ f(ξ) exp( 4π 2 κξ 2 t) exp ( 2πiξ(x + αt) ) ï dξ dy f(y) dy f(y)e 2πiξy ò exp( 4π 2 κξ 2 t) exp ( 2πiξ(x + αt) ) dξ e 4π2 κξ 2t e 2πiξ((x y)+αt) dy D((x + αt) y, t)f(y). (4.23) D(x, t) exp( 4π 2 κξ 2 t) Fourier D(x, t) = dξ e 4π2 κξ 2t e 2πiξx. (4.24) κ > 0 (4.24) t > 0 Fourier D(x, 0) = dx e 2πiξx = δ(x) (4.25) (4.24) D(x, t) D t = D κ 2 x 2 D(x + αt, t) (4.21) (4.23) u(x, 0) = dy D(x y, 0)f(y) = A = 4π 2 κt > 0, B = 2πx D(x, t) = dxi e Aξ2 ibξ dy δ(x y)f(y) = f(x) (4.26)

137 4.3 Fourier 131 Aξ = k, dξ = dk/ A, β = B/2 A D(x, t) = exp( B 2 4A ) dk e (k+iβ)2 A exp( z 2 ) ( R, R) Γ Cauchy Γ dz e z2 = 0. r dk e (k+iβ)2 = D(x, t) = exp( B 2 4A ) π = A dk e k2 = π 1 2 πκt e x 2 4κt (4.25) lim t πκt e x 2 4κt = δ(x) (4.21) (4.23) 1 u(x, t) = 2 πκt ã (x y + αt)2 dy exp Å f(y) (4.27) 4κt (4.26) f(x) u(x, 0) x + αt κt ) 4.6 (4.21) κ = 1, α = 0.3 f(x) = 50, x , x > 0.01 Mathematica

138 Green Fourier [14] ï d 2 dx 2 + p d ò dx + q f(x) = R(x) (4.28) Fourier F Å d 2 f F dx 2 + p df ã dx + qf = [ (iλ) 2 + p(iλ) + q ] f (4.28) f(x) = 1 2π = 1 2π = F(R) = R. (4.28) R(x) = δ(x) dλ e iλx f(λ) (4.29) R(λ) dλ e iλx λ 2 + ipλ + q. (4.30) (4.8) Fourier F(R) δ(λ) = (1/2π) 1/2 (4.30) ï d 2 dx 2 + p d ò dx + q G(x) = δ(x) (4.31) G(x) (2π) 1/2 Fourier Fourier (4.28) L(x) L(x)f(x) = R(x) (4.32) L(x)G(x) = δ(x) (4.33)

139 4.3 Fourier 133 G(x) (4.30) R = (2π) 1/2 L(x) Fourier Fourier G(x) L(x) Green Green (4.33) Green (4.32) f(x) = dx G(x x )R(x ) (4.34) (4.34) (4.32) dx [ L(x)G(x x ) ] R(x ) = = R(x) dx δ(x x )R(x ) (4.34) Green G(x) R(x) (convolution) f(x) = G R(x) 3.3 Fourier 3.25 Fourier F(G R) = 2πF(G)F(R) (4.32) F(f) = 2πF(G)F(Lf). (4.33) Green L(x) 4.7 ( ) d 2 x dx + 2β dt2 dt + ω2 0x = R(x), β, ω 0 > 0. (4.35) x(t) (4.34) Green G(x) ï d 2 dt 2 + 2β d dt + ω2 0 ò G(x) = δ(x).

140 134 4 G(x) Fourier G(ω) G(ω) = 1 2π = 1 2π 1 (ω 2 0 ω2 ) + i2βω d(t t ) e iω(t t ) G(t t ) Green G(t) = 1 2π dω e iωt G(ω) Fourier 3.17 G(t) = 1 ω 1 e βt sin ω 1 t θ(t). θ(t) θ(t) = ω 1 = 0, t < 0 1, t 1» ω 2 0 β2 x(t) x(t) = 1 ω 1 t dt e β(t t ) sin ω 1 (t t )R(t ) Green t t R(t ) R(t ) = e iωt (4.35) x(t) = e iωt (ω 0 Ω) 2 + i2βω (4.36) 4.8 (4.36) (4.35)

141 Schwartz Lighthill 4.3 L.Schwartzh D [7] 4.4 Dirac [11] f(x) f [a, b] f g f g L f g = b a dx f(x) g(x) w(x). (1.9 ) C n x, y i- x i y i n x y = x k y k (1.1 ) k=1 x- f(x) g(x) n- a {e i } i=1,...,n n a = a i e i, a i = e i a i=1 x f x- f(x) = x f (4.37)

142 136 4 f x- x x x x x x x x = 0 (4.38) 1.41 (1.31) f f = = b a b a dx w(x)f(x) x (4.39) dx x w(x) x f (4.40) x x Î = b a dx x w(x) x (4.41) f x x f = f(x ) = b a dx w(x)f(x) x x. (4.42) (4.38) x x?? x x = 1 w(x)w(x ) δ(x x ) = 1 w(x) δ(x x ) = 1 w(x ) δ(x x ) (4.39) f(x ) = b a dx f(x)δ(x x ) (4.43) Kronecker a i = k a k δ i,k f δ(x) Schwartz [7]

143 (4.41) Î x x δ(x x ) = b δ i,j = k δ i,kδ k,j a dx δ(x x)δ(x x ) (4.44) Dirac observable Herimite Ô o ϕ Ô o Ô ϕ = o ϕ x Hermite ˆx x x ˆx x = x x. ˆx x ˆx x x ˆx x = x x x x x x = x (4.38) x dx x x = 1 x Dirac [11, p.78,p.438 ]

144

145 139 5 Fourier x = (..., x 2, x 1, x 0, x 1, x 2,... ) l signal x(n) < n Z {x(n)} n Z = (..., x( 2), x( 1), x(0), x(1), x(2),... ) x n x(n) x x(n) l 2 n x(n) 2 < [a, b] f T [sec] f ν = 1/T [Hz] ω = 2πν 2.1 [a, b] N x = (b a)/n N 3.38 λ > Ω f(λ) = 0 2Ω f = F 1 [ f] T = πj/ω { f (πj/ω)} j Z }

146 140 5 Fourier f Nyquist 5.2 filter x y L Lx = y n Lx (n) = y (n). (1) L x 1 (n), x 2 (n) a, b L(ax 1 + bx 2 )(n) = alx 1 (n) + blx 2 (n). (2) L C x(n) Lx(n) C x(n). n Z n Z (3) k Z τ k τ k x(n) = x(n k). (4) LTI, liner translation invariant L L(τ k x) = τ k (Lx), ( ) L(τ k x)(n) = τ k (Lx)(n) = Lx(n k). (5) x 1 (n) x 2 (n) x 1 x 2 (n) y(n) = x 1 x 2 (n) = k Z x 1 (k)x 2 (n k). 5.3 (1) x 1 (n) x 2 (n) y(n) = x 1 x 2 (n) (2) x 1 (n) x 2 (n) (x 1 x 2 )(n) = (x 2 x 1 )(n). (3) h(n) L h x(n) = (x h)(n) LTI

147 x y L L e k, k Z 0 k n e k (n) = 1 k = n x(k) x = k Z x(k)ek L Lx = k Z L(x(k)e k ) = k Z x(k)l(e k ) Le k = E k τ p (E k ) = τ p (Le k )) = L(τ p e k ) L = L(e k+p ) = E k+p E k+p E k E k+p (n) = E k (n p) E k (n) = E 0 (n k) L x Lx(n) = x(k) L(e k )(n), E k (n) = E 0 (n k) }{{} k Z E k (n) = k Z x(k)h(n k), h E 0 = L(e 0 ) = x h(n) unit impulse signal 1 n = 0 δ(n) = 0

148 142 5 Fourier δ(n) 5.6 x(n) = x(k)δ(n k) = x δ(n) (5.1) k Z 5.4 LTI h 5.7 L LTI h(n) Lx(n) = (x h)(n) = x(k)h(n k) (5.2) k Z h(n) = (Lδ)(n) L h(n) n h(n) < x(n) (5.1) x(n) = k Z x(k)τ k δ(n) L LTI (Lx)(n) = k Z x(k)(lτ k δ)(n) = k Z x(k)τ k (Lδ)(n) = k Z x(k)h(n k) (5.2) L 5.8 LTI L 5.7 Lx(n) = (x h)(n) h(n) L inpulse response L system function h(n) ĥ(ω) 5.10 L 5.9 LTI L h(n) ĥ(ω) L x(n) = e i2πnω0 n x(n) = n 1 = x(n)

149 (T x)(n) = (x h)(n) = k Z x(k)h(n k) = k Z x(n k)h(k) = k Z e i2πω0(n k) h(k) = e i2πnω0 k Z = x(n)ĥ(ω 0). h(k)e i2πkω0 x(n) L ĥ(ω) ĥ(ω) LTI 5.10 x = {x(n)} n Z [ π, π] x(ω) = n Z x(n)e iωn X ( e iω) (5.3) z = e iω X(z) = n Z x(n)z n (5.4) Z- Z-tansform 5.11 Fourier f(ω) = 1 2π f(x) = 1 2π f(x)e i2πωx dx (?? ) f(ω)e i2πωx dω (?? ) x = {x(n)} x(ω) Fourier 5.3 n x(n) < x(ω) 1 x(ω) X(e iω ) {e iω } ω [0,1) X(z)

150 144 5 Fourier (a) (b) 5.1 (a) N = 10 x(0) = = x(9) = 1 x(ω) = ( 1 e iωn) / ( 1 e iω) (b) a = 0.8 < 1 x(n) = a n (0 n) x(n) = 0(n < 0) x(ω) = 1/ ( 1 ae iω) 5.12 (a) N x(n) x(n) = 1 0 n < N 0. x(ω) = X(z) = N 1 n=0 e iωn = 1 e iωn 1 e iω 5.1(a) i(n 1)ω/2 sin Nω/2 = e sin ω/2 N 1 n=0 z n = 1 z N 1 z 1.

151 5.2 l 2 (Z N ) 145 (b) x(n) x(n) = a n 0 n 0 n < 0. x(ω) = X(z) = ( e iω ) n 1 =, 5.1(b) 1 ae iω n=0 a n z 1 1 = 1 az 1. n= x 1 (n) x 2 (n) y(n) = (x 1 x 2 )(n) ŷ(ω) = x 1 (ω) x 2 (ω) ŷ(ω) = y(n)e iωn n Z = x 1 (k)x 2 (n k)e iωn n Z k Z = x 1 (k) x 2 (n k)e iωn n Z k Z = x 1 (k)e iωk x 2 (n k)e i(n k)ω = x 1 (ω) x 2 (ω) n Z k Z 5.2 l 2 (Z N ) N x(0), x(1),..., x(n 1) x(n mod N), n Z N x( 1) = x(n 1), x(n) = x(0) N = 12 x( 21) = x( 9) = x(3) = x(15) = x(27)

152 146 5 Fourier 5.14 N N {x(n)} n Z N x(n + N) = x(n), n Z N S N N N N Z mod N Z N = {0, 1,..., N 1} N x(n) S N, x(n + N) = x(n) Z N x = (x(0), x(1),..., x(n 1)) 1.17 l 2 x y l 2 = N 1 n=0 x ny n (1.18 ) l 2 (Z N ) l N E 0, E 1,..., E N 1 l 2 (Z N ) E k = {E k (n)} n=0,...,n 1, E k (n) = 1 N e i2πkn/n 0 k N 1 (5.5) E k n- E k (n) 5.16 { E 0, E 1,..., E N 1} l 2 (Z N )

153 5.2 l 2 (Z N ) 147 E j 0 k, n N 1 E k N 1 = n=0 N 1 E j (n)e k (n) = N 1 n=0 = 1 e i2π(k j)n/n = 1 N N n=0 n=0 1 k = j = 1 1 e i2π(k j) N 1 e = 0 k j i2π(k j)/n = δ jk 1 e i2πjn/n 1 e i2πkn/n N N N 1 Ä e i2π(k j)/n ä n 5.17 N = 4 E 0 = 1 (1, 1, 1, 1), 2 E 1 = 1 (1, i, 1, i), 2 E 2 = 1 (1, 1, 1, 1), 2 E 3 = 1 (1, i, 1, i) l 2 (Z N ) l 2 (Z N ) I l 2 (Z N ) = N 1 k=0 E k E k x l 2 (Z N ) x = I l2 (Z N ) x x = {x(n)} n=0,...,n 1 = N 1 k=0 N 1 = 1 N E k x E k (n) k=0 x(k) e i2πkn/n (5.6) (5.6) x(k) Fourier x(k) = E k x = N 1 n=0 x(n) e i2πkn/n (5.7)

154 148 5 Fourier (5.6) Fourier {E k } k=0,...,n 1 x Fourier 5.3 { x = { x(k)} k=0,...,n 1 x N- Fourier Fourier Fourier N { x(k)} k=0,...,n 1 x(t) [0, 2a] N x(0), x(1),..., x(n 1) x(t) k Fourier γ k γ k 1 x(k) (5.8) N 5.2 [0, 2a] x(t) 1 x(t)dt N 0 [0, 2a] N t 0 = 0, t 1 = 2a/N, t 2 = 2 2a/N,..., t k = 2ka/N,..., t N = 2a [t j, t j+1], 0 j < N 2a 2a 2a x(t), x(t + 2a) = x(t) Fourier x(t) Fourier (2.25) x(t) = γ k e iπkt/a (2.28 ) k=

155 5.3 Fourier γ k = 1 2a 2a 0 x(t)e iπkt/a dt (2.29 ) 2a [ ] 5.2 x(t) [0, 2a] 1 2a N x(t)dt 0 N N t = 2a/N N t 0 = 0, t 1 = 2a/N, t 2 = 2 2a/N,..., t k = 2ka/N,..., t N = 2a N x(t j ) = x(2aj/n) x(j), j = 0,..., N [t j, t j+1 ], 0 j < N 1 2a 2a 0 x(t)dt 1 ï x(0) 2a N t 2 = 1 N + x(1) + + x(n 1) + x(n) ò 2 x(t) 2a x(0) = x(n) N 1 j=0 x(j). 2a x(t) Fourier k- Fourier γ k N { x(j) = x(2aj/n) } j=0,...,n 1 γ k = 1 2a 1 N 2a 0 N 1 x j=0 x(t)e iπkt/a dt Å ã 2aj e i2πkj/n N (5.7) Fourier x(k) γ k 1 N x(k) (5.8 ) 5.3 Fourier 5.2 S N N Z N 5.1 n Z

156 150 5 Fourier x(n) M M LTI 5.2(c)-(e) 5.8 LTI T x(n) = (x h)(n) 5.18 h N x S N x h N x h(n) = k Z x(k)h(n k) n n x(n) x(n) M x h(n) = k Z x(k) h(n k) M k Z h(n k) = M k Z h(k) <. x h(n) N x h(n + N) = k Z x(k)h(n + N k) = k Z x(k)h(n (k N)) = k Z x(k + N)h(n k) = k Z x(k)h(n k) = x h(n) ( Fourier ) N x S N x N- Fourier DFT x N x(n) = N 1 k=0 x x F N [x] = x x(k)e i2πkn/n (5.9)

157 5.3 Fourier F N N N S N F N x S N N F N [x] S N F N [x] N ω = e i2π/n x(k + N) = = = N 1 n=0 N 1 n=0 N 1 n=0 = x(k). x(n)ω n(k+n) x(n)ω nk ω Nn x(n)ω nk omega Nn = e i2πn = 1 x N ( Fourier ) N x S N DFT F N [x] = x x(n) = 1 N N 1 k=0 x x F 1 N x = F 1 N [ x] x(k)e i2πkn/n, n Z. (5.10) r N 1 k=0 N 1 k=0 r k = 1 rn 1 r î e i2π(n j)/n ó k = 1 e i2π(n j) 1 e i2π(n j)/n.

158 152 5 Fourier 0 n N N 1 k=0 0 n j e i2πk(n j)/n = N n = j 0 n N 1 (5.11) N 1 1 x(k)e i2πkn/n = 1 N N k=0 = N 1 j=0 N 1 = x(n). N 1 k=0 j=0 x(j) 1 N x(j)e i2πjk/n e i2πkn/n N 1 k=0 e i2πk(n j)/n 5.19 N- Fourier (5.9) x(n) N- DFT x(n) x(0) x(0) x(1) x =, x = x(1).. x(n 1) ω N = e i2π/n. x(n 1) N N F N F N = [ ω jk ] N ω N ωn 2... ω N 1 N = 1 ωn 2 ωn 4... ω 2(N 1) N ω N 1 N ω 2(N 1) N... ω (N 1)2 N FFourier (5.9) x(n) = ( F N x ) (n) j,k=0,...,n 1 (5.12) (5.13)

159 5.3 Fourier (5.12) F N F N N F N = [f 0 f 1... f N 1 ] f j f k = Nδ jk (5.11) Fourier y = F 1 N [ŷ] (5.10) F N F N x(n) = 1 ( F N N x) (n) = 1 ( F N N F Nx ) (n) N N I N I N = Å F N N ã F N N U N = F N N 5.15 E k U N k 5.23 (DFT ) S N N DFT (1) N x S N 1 y(n) = x(n + 1), n Z y S N ŷ(k) = e i2πk/n x(k). (2) N x, y S N circular convolution x y(n) = N 1 k=0 x(k)y(n k) (5.14) x y S N (5.14) N (3) N x, y S N DFT x, ŷ x y(n) DFT x y(n) x y(n) = x(n) ŷ(n).

160 154 5 Fourier (4) x S N DFT x x x(n k) = x (k), 0 k N 1. N DFT x Z N x(0),..., x(n/2 1 (1) 1 y(k) = x(k + 1) ŷ(k) = N 1 j=0 y(j)e i2πkj/n = N 1 = e i2πk/n j=0 = e i2πk/n x(k). N 1 j=0 x(j + 1)e i2πk(j+1)/n x(j + 1)e i2πk/n e i2πk(j+1)/n (2) m (m 1)N j < mn N+j 1 k=j x(k)y(n k) = = = = mn 1 k=j mn+j 1 x(k)y(n k) + mn (m 1)N 1 k=j (m 1)N N 1 k=j (m 1)N N 1 k=0 x(k)y(n k) k=mn x(k)y(n k) N+j 1 mn x(k)y(n k) + k=0 x(k)y(n k) j (m 1)N 1 x(k)y(n k) + x(k)y(n k) k=0 = N 1 k=0 x(n k)y(k) = y x(n)

161 5.3 Fourier 155 (3) S N N 1 x y(n) = (x y)(k)e i2πnk/n = = k=0 N 1 N 1 k=0 j=0 N 1 N 1 k=0 j=0 x(j)y(k j)e i2πnk/n x(j)e i2πnj/n y(k j)e i2πn(k j)/n = N 1 x(j)e i2πnj/n y(k j)e i2πn(k j)/n j Z k=0 = x(n)ŷ(n). (4) x S N x(n k) = = N 1 j=0 N 1 j=0 = x (j) x(j)e i2π(n k)j/n x(j)e i2πkj/n e i2π = 1

162 156 5 Fourier (a) (b) (c) 5.3 (a) f(x) = e x/3( sin 3x + 2 sin 5x sin 2x sin 40x ) [0, π] 2 7 = 128- y f(x) 3.6(a) (b) y DFT ŷ (c) K = 19 ŷ(0)..., ŷ(k 1) K ŷ(n K)..., ŷ(n 1) ŷ K (c) (c) ŷ ŷ Z N=128 ŷ(n k) = ŷ (k), k = 0,..., N 1 ŷ ŷ(0),..., ŷ(n/2 1) Butterworth 3.35 [0, π] f(x) = e x/3( sin 3x + 2 sin 5x sin 2x sin 40x ) (3.25) 2 7 = 128 y = { } f(πk/n) 5.3(a) k=0,...,n 1 y DFT ŷ ŷ(k) N k f k- Fourier γ k Riemann-Lebesgue 2.32 Fourier k = N/2 ŷ(k) 5.23(4) ŷ(n k) = ŷ (k) 5.3(c) ŷ k = 1 ŷ (N 1) = ŷ(1) ŷ(n 1) γ N 1

163 5.3 Fourier (b) ŷ K = 19 ŷ(0),..., ŷ(k) K ŷ(n K),..., ŷ(n 1) ŷ K=19 Fourier 2K/N ( ) K = 10 Fourier Fourier ŷ 15% 117 Fourier 5.4(b) 8.6% 1 117/128) f (a) (b) 5.4 (a) y DFT ŷ K = 10 ŷ(0)..., ŷ(k) K ŷ(n K)..., ŷ(n 1) ŷ K=10 (b) ŷ 15% Mathematica 5.3(a),(b) (c) data m n Mathematica 1 0 data[[m;;n]] 5.26 Mathematica 5.4(b) data Fourier Fourier

164 158 5 Fourier Fourier[data] Fourier Chop ReplaceAll /. Condition /; data value A data /. x_ /; Abs[x] < value -> A; 5.27 h(n) x(n) N x(n) x(n) DFT ĥ(ω) 5.10 x h(n) = x(n)ĥ(n/n). N 1 x h(n) = (x h)(k)e i2πnk/n = = k=0 N 1 x(j)h(k j)e i2πnk/n k=0 j Z N 1 x(k j)h(j)e i2πnk/n k=0 j Z = j Z N 1 h(j)e i2πj(n/n) = ĥ(n/n) x(n). k=0 x(k j)e i2πn(k j)/n 5.28 ĥ(ω) 1 ĥ(ω + 1) = ĥ(ω) n+n n N ĥ( N ) = ĥ( n N ) ĥ(n/n) N h(n) N- DFT h(n) 5.21 ĥ(n/n) DFT

165 5.4 Fourier 159 h(n) = 1 N = 1 N N 1 k=0 N 1 k=0 j Z = h(j) 1 N j Z N 1 k=0 ĥ(n/n)e i2πkn/n h(j)e i2πjk/n e i2πkn/n N 1 k=0 e i2πk(n j)/n = = m Z h(n + mn). e i2πk(n j)/n N j n = mn, m Z 0 j n mn h(n) h(n) N- h(n) = m Z h(n + mn) 5.4 Fourier Mathematica FFT( Fourier ) Fourier Mathematica data data/yushima.jpg 1000 data/syushima.jpg gray SetDirectory[NotebookDirectory[]] yushima = Import["data/Yushima.jpg"] Export["data/sYushima.jpg", yushima, ImageSize -> 1000] gray = ColorConvert[Import["data/sYushima.jpg"], "Grayscale"]

166 160 5 Fourier ImageData[gray] takedata 5.5 takedata = Take[ImageData[gray], {451, 750}, {151, 550}] Image[takedata] Image[takedata] takedata FFT 5.6 Fourier FFT InverseFourier Mathematica Chop ListPlot[takedata[[150]], Filling -> Axis] fft = Fourier[takedata[[150]]]; ListPlot[Abs[fft], AxesLabel -> {k,}, Filling -> Axis] Mathematica FFT Fourier, InverseFourier FFT Fourier 2 FFT approx2ddatabyneglectingsmallelements[data2d,thresholdratio]

167 5.4 Fourier (1) (2) k (3) (1) (2) FFT (3) FFT Chop (1) 2 data2d 1 thresholdratio FFT maxabs maxabs thresholdratio Ignoring rate Ignoring rate approx2ddatabyneglectingsmallelements[data2d_,thresholdratio_] := Module[{k, fftlist, ignorfftlist, replaced = 0}, ModifiedListIngoringRelativeSmallElements[lst_List] := Module[{maxabs, approxed, replacesymbol}, maxabs = Max[Abs[lst]]; approxed = lst /. x_ /; Abs[x] < thresholdratio*maxabs -> replacesymbol;

168 162 5 Fourier replaced += Count[approxed, replacesymbol]; approxed /. replacesymbol -> 0 ]; fftlist = Fourier[data2D]; ignorfftlist = Table[ ModifiedListIngoringRelativeSmallElements[fftlist[[k]]], {k, 1, Length[data2D]}]; Print["Ignoring rate=", replaced / Length[Flatten[data2D]] // N]; Chop[InverseFourier[ignorfftlist]] ] 2 takedata 3% Image[approx2DdataByNeglectingSmallElements[takedata, 0.03]] Ignoring rate= (1) (4) (1) 0.03 (2) 0.04 (3) 0.06 (4) 0.08

169 5.4 Fourier 163 (1) (2) (3) (4) 5.7 FFT (1) 0.03 Ignoring rate= (2) 0.04 Ignoring rate= (3) 0.06 Ignoring rate= (4) 0.08 Ignoring rate=

170

171 II Wavelet

172

173 167 6 Haar 1.29 Haar ϕ Haar wavelet ψ 6.1 Haar Haar dyadic 6.1 j k dyadic I j,k I j,k = [ 2 j k, 2 j (k + 1) ) (6.1) 6.2 j 2 j dyadic I j,k R R = k Z I j,k dyadic 6.3 j 0, k 0, j 1 j 0, k 1 k 1 Z (1) (2) (1) I ji,k 1 I j0,k 0 = ϕ, (2) I j1,k 1 I j0,k 0 I j0,k 0 I j1,k j dyadic I j,k I j,k = I l j,k I r j,k

174 168 6 Haar Ij,k l = I j+1,2k, Ij,k r = I j+1,2k j diadic f f dyadic supf = I j,k k=k 0,...,k n 1 I j,k f(x) =, x I j,k 6.6 f(x) j dyadic f j (j j) dyadic Haar 1.29 Haar 6.7 (Haar ) ϕ(x) = χ [0,1) (x) = 1 0 x < 1 0 (6.2) ϕ(x) dyadic I 0,0 x = 0, 1 ϕ(x k) I 0,k x = k, k + 1 ϕ(mx) dyadic I 0,m V 0 0 diadic { } V 0 = cl c k ϕ(x k) a k R, Λ = span {ϕ(x k)} k Λ. k Λ ϕ(2x) + 2ϕ(2x 1) + 2ϕ(2x 2) ϕ(2x 3)

175 6.1 Haar Haar Haar ϕ(x) ϕ(x ) ϕ(x) ϕ(x ) L 2 = ϕ(t) ϕ(t ) dt 6.12 {ϕ(x k)} k Z V V j 0 dyadic { } V j = cl c k ϕ(2 j x k) c k R, Λ = span{ϕ(2 j x k)} k Λ k Λ V j j approximate space 6.2 Haar 6.3 Haar V j 6.14 j Haar ϕ j,k dyadic I j,k ϕ j,k = 1 ϕ j,k (x) = 2 j/2 ϕ(2 j x k) (6.3) 6.15 j Haar {ϕ j,k (x)} k Z V j ϕ j,k (x) ϕ j,l (x) L 2 = 2 j/2 ϕ(2 j x k) = 2 j = δ kl 2 j/2 ϕ(2 j x l) L 2 ϕ(2 j x k) ϕ(2 j x l) dx = 2 j 1 2 j δ kl V 0 V j f(2 j x) V j f(x) V 0,

176 170 6 Haar f(2 j x) V 0 f(x) V j f(x) V 0 f(x) = k c kϕ 0,k (x) f(2 j x) = k c kϕ 0,k (2 j x) = k c kϕ j,k (x) f(2 j x) V j f(x) V j f(x) = k c k ϕ j,k(x) f(2 j x) = k c k ϕ j,k(2 j x) = k c k ϕ 0,k(x) f(2 j x) V V 0, V 1,..., V j,... V 0 V 1 V j V j f V j V j+1 V j V j+1. strict V j V j+1 V j+1 V j f dyadic I j+1,0 f(x) = ϕ(2 j+1 x) V j+1 V j dyadic I j,k ϕ(2 j x k) ϕ(2 j+1 x) f V j V j+1 V j V j+1 \V j Haar wavelet 6.17 j Haar V j V j 1 f V j f V j 1 V j V j = V j 1 W j 1 W j 1 j 1 detail space W j 1 V j V j 1 V j 1 j = 1 ψ (1) ψ V 1 ψ(x) = l c lϕ(2x l), c l R, (2) ψ V 0 k ψ(t) ϕ(t k)dt = 0.

177 6.1 Haar Haar 171 (1) dyadic 1/2 (2) ϕ 0 ψ ψ(x) = ϕ(2x) ϕ(2x 1) ψ ϕ L 2 = ψ(t) ϕ(t) dt = 1/2 0 1 dt dt = 1/2 1/2 = 0 1/2 ψ(x) I 0, ϕ(x k), k 0 I 0,k k 0 ψ(x k) ϕ(x)dx = 0 V 1 ψ V 0 ψ V0 ψ (Haar wavelet ) ψ(x) = ϕ(2x) ϕ(2x 1) (6.4) 6.19 ψ(x 1), ψ(x 2), ψ(x 3) ψ(2x 1), ψ(2x 2), ψ(2x 3) 6.20 ϕ(2x) ϕ(x k) ψ(x l) 6.28 ϕ(x k) ϕ(2x) L 2 ϕ(x k) + ψ(x k) ϕ(2x) L 2 ψ(x k) k = ϕ(x) ϕ(2x) L 2 ϕ(x) + ψ(x) ϕ(2x) L 2 ψ(x) = 1 2 ϕ(x) ψ(x) 6.21 f(x) = 2ϕ(4x) + 2ϕ(4x 1) + ϕ(4x 2) ϕ(4x 3) f(x) = ψ(2x 1) + 2ϕ(2x) f(x) = ψ(2x 1) + ψ(x) + ϕ(x) 6.22 j Haar wavelet ϕ j,k ψ j,k (x) = 2 j/2 ψ(2 j x k) (6.5) 6.23 ψ j,k (x) j + 1 dyadic ψ j,k (x) = 2 j/2 ( χ Ij+1,2k (x) χ Ij+1,2k+1 (x) )

178 172 6 Haar ψ j,k ψ j,k L 2 = δ kk 6.2 Haar dyadic I j,k Haar ϕ j,k Haar wavelet ψ j,k Haar Haar system Haar 6.24 V 1 f 1 (x) = k c kϕ(2x k) f 1 V 0, l f 1 ϕ(x l) c 1 = c 0, c 3 = c 2,..., c 2l+1 = c l, V 1 f 1 (x) = k c k ϕ(2x k) ϕ(2x k) dyadic I 1,k ϕ(x l) I 0,l f 1 (x) ϕ(x l) L 2 = Å c k ϕ 2(t k ã k 2 ) ϕ(t l) dt = l c 2l l+ 1 2 l + c 2l+1 l+1 l+ 1 2 ϕ (2(t l)) ϕ(t l) dt Å ϕ 2(t l 1 ã 2 ) ϕ(t l) dt = 0 c 2l = c 2l+1 f 1 (x) = l = l c 2l ϕ(2x l) + c 2l+1 ϕ(2x l 1) = l c 2l ψ(x l) V 0 = W 0 c 2l [ϕ(2x l) ϕ(2x l 1)]

179 6.2 Haar 173 V 1 f 1 (x) V 0 f 1 f 1 (x) = l Z c 2l ψ(x l) W 0 = V 0 V 1 V 0 c 2k c k k c kψ(x k) { } W 0 = cl k Λ c k ψ 0,k (x) a k R, Λ = span{ψ 0,k (x)} k Λ V 1 V 0 V 0 = W 0 V 1 = V 0 W 0 W j V j W j Haar wavelet {ϕ j,k (x)} k {ψ j,k (x)} k { } V j = cl k Λ { c k ϕ j,k (x) a k R, Λ = span{ϕ j,k (x)} k Λ, (6.6) } W j = cl d k ψ j,k (x) b k R, Λ = span{ψ j,k (x)} k Λ k Λ (6.7) W j V j+1 V j V j W j, V j+1 = V j W j. 1) W j V j 2) V j V j+1 W j 1) W j g(x) = k d k ψ j,k (x) f V j g f L 2 = g(t) f(t) dt = k ( 2 j/2 d k ψ(2 j t k)) f(t) dt = 0

180 174 6 Haar f(x) V j 6.16 f(2 j x) V 0 0 = ( d k ψ(t k)) f(2 j t) dt ψ(x) V 0 k = 2 j d kψ(2 j y k) f(y) dy y = 2 j t k = 2 j/2 g(y) f(y) dy. g f V j 1) 2) j = j 6.1 J dyadic f J J 1 dyadic f J 1 dyiadic I J 1,k = I J,2k I J,2k+1 f J 1(x) f J(x) I J 1,k c J(2k) c J(2k +1) 1 (cj(2k)+cj(2k +1)) V j

181 6.2 Haar V j = W j 1 V j 1 = W j 1 W j 2 V j 2... = W j 1 W j 2 W J V J... = W j 1 W j 2 W 0 V 0 (6.8) 6.27 MRA: Multi Resolution Analysis Fourier Haar MRA MRA Wavelet ( ) f J (x) J dyadic f J (x) f J (x) = w J 1 (x) + f J 1 (x), f J 1 (x) J 1 approximated function w J 1 (x) J 1 detail function J 1 Haar wavelet w J 1 = k d J 1 k ψ J 1,k (x), (6.9) f J (x) J dyadic f J (x) I J,k c J (k) f J (x) = k c J k χ IJ,k (x), c J k = 2 j I j,k f(t) dt I J 1,k = I J,2k I J,2k+1 J 1

182 176 6 Haar f J 1 (x) ô f J 1 (x) = ñ2 J 1 f J (t)dt χ IJ 1,k (x) I J 1,k Ç å ô = ñ2 J 1 f J (t)dt χ IJ 1,k (x) = 1 2 I J,2k + I J,2k+1 ( c J 2k + c J ) 2k+1 χij 1,k (x). (6.10) 6.1 I J 1,k f J 1,k (x) I J 1,k f J (x) w J 1 (x) = f J (x) f J 1 (x) 6.6 w J 1 J dyadic w J 1 (t)dt = f J (t)dt f J 1 (t)dt I J 1,k I J 1,k I J 1,k Ç å = f J (t)dt + f J (t)dt f J 1 (t)dt I J,2k I J,2k+1 I J 1,k = ( 2 J c J 2k + 2 J c J ) 2k+1 2 (J 1) 1 ( c J 2 2k + c J 2k+1) ) = 0 I J 1,k w J 1 (x) Haar wavelet ψ J 1,k (x) (6.9) f J (x) = w J 1 (x) + f J 1 (x) 2 J f J (x) 2 I J 1,k f J 1 (x) I J 1,k I J 1,k = I j,2k I j,2k+1 w J 1 (x) J 1 w J V J f J j(j < J) V j j W J 1, W J 2,..., W j f J f J (x) = w J 1 (x) + w J 2 (x) + + w j (x) + f j (x)... = w J 1 (x) + w J 2 (x) + + w 0 (x) + f 0 (x)

183 6.2 Haar 177 w n f j Haar wavelet {ψ n,k } W n f J f j Haar {ϕ J,k } V J Haar Haar decomposition Haar Haar Wavelet transformation Wavelet : χ [0,1/2) ϕ j,k ψ j,k f(x) = χ [0,3/4) (x) [0, 3/4) dyadic [0, 3 4 ) = I 1,0 I 1,1 I 0,0 = [0, 1) f(x) = 0 [0, 1) Dyadic ϕ j,k ψ j,k f ϕ 0,0 L 2 = 3/4 k 0 f ϕ 0,k L 2 = 0. I j,k [0, 3/4) I j,k [3/4, 1) j, k 2 j 0 k 2 j 1 f ψ j,k L 2 = 0. f ψ 0,0 L 2 = 1/4 f ψ 1,1 L 2 = 2 3/2 χ [0,3/4) (x) = 2 3/2 ψ 1,1 (x) ψ 0,0(x) ϕ 0,0(x) f(x) = χ [0,11/16) (x) Haar 6.31 f(x) = χ [0,2/3) (x) Haar f ϕ 0,0 L 2 = 2/3 j, k 0 f ϕ 0,k L 2 = 0. [0, 2/3) dyadic j 0 ψ j,k (x) f ψ j,k L 2 j 0 2/3 k j dyadic I j,kj χ [0,2/3) (x) = c j (k j )ψ j,kj (x) ϕ 0,0(x) j=0 = c j k j ψ(2 j x k j ) ϕ(x), cj k j = 2 j/2 c j (k j ) j= f(x) = χ [0,2/3) (x) Haar c j (k j ) c j (k j) j = 0, 1, 2, 3, 4, 5 {V j } {W J }

184 178 6 Haar 6.33 (Haar ) (1) W J J Haar wavelet {ψ J,k (x)} k Z W J (2) V J W J J {ϕ J,k, ψ J,k } k,k Z R V J W J. (3) {W j } j Haar wavelet {ψ j,k (x)} k Z R j j W j W j. (4) J Haar {ϕ J,k, ψ j,k (x)} j J,k,k Z R V J Haar J wavelet j J J j 1) J k, k Z dyadic I J,k, I J,k I J,k I J,k = 0, k k ψ J,k ψ J,k L 2 = I J,k k = k R R ψ J,k (t) ψ J,k (t) dt = 0 k k ψ J,k (t) 2 dt = 1 k = k. 2) ) k, k Z, j > j (1) I j,k I j,k = ϕ ψ j,k ψ j,k L 2 = 0. (2) I j,k Ij l,k Il j,k ψ j,k = 1. ψ j,k ψ j,k L = 2 ψ j,k (t) ψ j,k (t) dt = I j,k ψ j,k (t)dt = 0. I j,k

185 6.3 Haar 179 (3) I j,k Ij r,k Ir j,k ψ j,k = 1. ψ j,k ψ j,k L = 2 ψ j,k (t) ψ j,k (t) dt = I j,k ψ j,k (t)dt = 0. I j,k (4) j J ϕ J,k (x) ψ j,k (x) diadic ) k, k Z, j J ϕ J,k ψ j,k = ϕ J,k (t) ψ j,k (t) dt = 0. R 6.3 Haar V j 6.25 Fourier Haar 6.34 j Z V j = span {ϕ j,k (x)} k Z f P j P j f(x) = k ϕ j,k f L 2 ϕ j,k (x) (6.11) P j f(x) f j Haar 6.35 {ϕ j,k (x)} R dyadic P j f(x) L 2 f(x) V j ϕ j,k (x) = 2 j/2 χ Ij,k (x) ϕ j,k f L 2 ϕ j,k (x) = 2 j ϕ(2 j x k) f L 2 ϕ(2 j x k) å = 2 Ç I j f(t)dt χ Ij,k (x) j,k P j f(x) dyadic I j,k V j j P j f(x) dyadic I j,k

186 180 6 Haar 2 j f(x) ϕ j,k f L 2 f P j 6.36 (1) P j f(x), g(x) α, β C P j (αf + βg)(x) = αp j f(x) + βp j g(x). (2) P j idenpotent Pj 2 = P j P j (P j f)(x) = P j f(x). (3) j j P j f V j P j f(x) = P j f(x). (4) P j f L 2 f L 2. (1), (2) P j (3) (2) g V j P j g = g 6.6 (4) {ϕ j,k (x)} k Z 2 P j f L 2 = f ϕ R j,k L 2 ϕ j,k (t) dt k = f ϕ j,k L 2 2 = 2 2j/2 f(t)dt. k k I j,k Cauchy-Schwarz 2 Ç å Ç å 2j/2 f(t)dt 2 j dt f(t) 2 dt = f(t) 2 dt. I j,k I j,k I j,k I j,k P j f L 2 f(t) 2 dt = I j,k R f(t) 2 dt = f 2 L 2.

187 6.3 Haar R f (1) (2) lim P jf f L 2 = 0, j lim P jf L 2 = 0. j (1) f(x) 2 N I J g V J f g = max x f(x) g(x) < ε 2 N+2 j > J 6.36(3) P j g(x) = g(x) Minkowski 6.36(4) P j f f L 2 P j f P j g L 2 + P j g g L 2 + g f L 2 = P j (f g) L 2 + g f L 2 2 g f L 2 g f 2 L = g(t) f(t) 2 dt < 2 g f L 2 < ε 2 I P j f f L 2 < 2 g f L 2 < ε. I ε 2 ε2 dt = 2N+2 4 (2) f(x) I I I j,k Dyadic I j,k P j f(x) = 2 j Ç I j,k f(t)dt å χ I j,k (x) P j f 2 L2 0, (j ).

188 182 6 Haar Haar 6.38 R f j Haar Q j Q j f(x) = P j+1 f(x) P j f(x). (6.12) Q j f 6.39 R f Q j f(x) Q j f(x) = k ψ j,k f L 2 ψ j,k (x). (6.13) j dyadic x I j,k P j f(x) = 2 j I j,k f(t)dt. I j,k j + 1 I l j,k = I j+1,2k I r j,k = I j+1,2k+1 I j,l = Ij,k l Ir j,k 2 j+1 f(t)dt, x I I P j+1 f(x) = l j,k l, j,k 2 j+1 f(t)dt, x I I r j,k r j,k 2 j/2, x Ij,k l, ψ j,k (x) = 2 j/2, x Ij,k r, 0, x Ij,k l Q j f(x) = P j+1 f(x) P j f(x) = 2 (2 j f(t)dt I l j,k ( = 2 j f(t)dt I l j,k = 2 j/2 ψ j,k f. I r j,k I l j,k f(t)dt ) f(t)dt I r j,k ) f(t)dt

189 6.3 Haar 183 x I r j,k ) Q j f(x) = 2 ( j f(t)dt + f(t)dt I l I r j,k j,k = 2 j/2 ψ j,k f I j,k Q j f(x) = ψ j,k f L 2 ψ j,k (x) j Q j 6.40 (1) Q j f(x), g(x) α, β C Q j (αf + βg)(x) = αq j f(x) + βq j g(x). (2) Q j idenpotent Q 2 j = Q j Q j (Q j f)(x) = Q j f(x). (3) j j Q j f W j Q j g(x) = 0. (4) Q j f L 2 f L W j ψ j,k 6.33 Haar P j Q j 6.41 (Haar ) J Haar {ϕ J,k (x), ψ j,k (x) j J} k Z R

190 184 6 Haar 6.33 (4) R f f(x) span{ϕ J,k, ψ j,k } j J,k Z ε > 0 N P N f f L 2 < ε N > J (6.12) N 1 j=j Q j f(x) = N 1 j=j P j+1 f(x) P j f(x) = P N f(x) P J f(x) (6.13) P N f(x) = N 1 j=j k Λ ψ j,k f ψ j,k (x) + k Λ ϕ j,k f ϕ j,k (x) f Λ P N f(x) span{ϕ J,k, ψ j,k } j J,k Z P N f f L 2 < ε 6.42 Haar {ψ j,k (x)} j,k Z R 6.33 (3) R f f(x) span{ψ j,k } j,k Z J N (6.12) J 1 j= J Q j f(x) = J 1 j= J P j+1 f(x) P j f(x) = P J f(x) P J f(x)

191 6.3 Haar lim P Jf f L 2 + P J f L 2 = 0 J Minkowski J 1 lim J f Q j f j= J L 2 = lim J f P Jf + P J f L 2 lim J P Jf f L 2 + P J f L 2 = 0. f Λ J 1 j= J Q j f(x) = J 1 j= J J 1 j= J k Λ ψ j,k f ψ j,k (x) Q j f(x) span{ψ j,k } j,k Z R L 2 f f(x) = j ψ j,k f ψ j,k (x) k f(x)dx = ψ j,k f ψ j,k (x)dx R R = j j k ψ j,k f k R ψ j,k (x)dx = 0 R L 2 f f(x)dx = 0 R f

192 186 6 Haar 6.44 L 2 (R) L 2 (R) = V 0 W 0 W f L 2 (R) f = f 0 + j=0 w j 6.4 Wavelet 6.28 J P J f(x) 2 j (j < J) wavelet decomposition wavelet transformation P J f(x) = Q J 1 f(x) + Q J 2 f(x) + + Q j f(x) + P j f(x) = Q J 1 f(x) + Q J 2 f(x) + + Q 0 f(x) + P 0 f(x) (6.14) P j f P j f(x) = k = k ϕ j,k f L 2 ϕ j,k (x) 2 j ϕ(2 j x k) f ϕ(2 j x k) = k å Ç2 j f(t) dt ϕ(2 j x k) I j,k (6.15) 2 j (j < J) J f(x) 6.37 P J f(x) ϕ J,k f L 2 = 2 J I J,k f(x)dx (6.14) Haar [a, b) f [a, b] [0, 1) τ : x x a b a

193 6.4 Wavelet j dyadic I j,k f 2 j I j,k f(x)dx f( k 2 j ) f( k+1 2 j ). f I j,k f( k 2 j ) < 2 j I j,k f(x)dx < f( k+1 2 j ) f [a, b) [a, b) τ τ [0, 1) f [0.1) (6.16) [a, b) f [0, 1) f = τ f τ 1 [a, b) [0, 1) f x =..., 2/2 j, 1/2 j, 0, 1/2 j, 2/2 j,... 2 j dyadic {I j,k } k Z {f ( k 2 j ) }k x = 1 2 j x f Shannon-Whittaker 3.38 f 2 Nyquist f = (f 0 f 1... f N 1 ) f f(x) [0, 1) f(x) (6.15) P j f(x) 2 j 2 j {c j k } k=0,...,2 j 1

194 188 6 Haar f j (x) 2 j f P j f P j f(x) f j (x) = 2 j 1 k=0 c j k ϕ(2j x k) = 2 j 1 k=0 Å ã k f 2 j ϕ(2 j x k) Haar ϕ wavelet ψ ψ(x) + ϕ(x) ϕ(2x) = 2 ϕ(x) ψ(x) ϕ(2x 1) = ϕ(2 j x) = ψ(2j 1 x) + ϕ(2 j 1 x) 2 ϕ(2 j x 1) = ϕ(2j 1 x) ψ(2 j 1 x) 2 (6.17) (6.18) ϕ(2 j x 2k) = ψ(2j 1 x k) + ϕ(2 j 1 x k) 2 ϕ(2 j x 2k 1) = ψ(2j 1 x k) ϕ(2 j 1 x k) 2 (6.17 ) (6.18 ) 6.28 Haar f(x) = 2ϕ(4x) + 2ϕ(4x 1) + ϕ(4x 2) ϕ(4x 3) Haar f(x) = ψ(2x 1) + ψ(x) + ϕ(x) 6.47 (Haar ) j f j (x) = k c j k ϕ(2j k) V j

195 6.4 Wavelet 189 f j 6.28 f j (x) = w j 1 (x) + f j 1 (x), w j 1 (x) = k f j 1 (x) = k d j 1 k ψ(2 j 1 x k) W j 1, c j 1 k ϕ(2 j 1 x k) V j 1 c j 1 k = cj 2k + cj 2k+1, (6.19) 2 k = cj 2k cj 2k+1. (6.20) 2 d j 1 f j = k = k c j k ϕ(2j k) c j 2k ϕ(2j x 2k) + k c j 2k+1 ϕ(2j x 2k 1) 6.45 f j (x) = c j ψ(2 j 1 x k) + ϕ(2 j 1 x k) 2k 2 Ç c j 2k å cj 2k+1 ψ(2 j 1 x k) + 2 = k = w j 1 (x) + f j 1 (x). c j ϕ(2 j 1 x k) ψ(2 j 1 x k) 2k k Ç c j 2k + å cj 2k+1 ϕ(2 j 1 x k) 2 f J 2 J {c J k = f ( k 2 J ) }k f J (x) 6.47 Haar f J = w J 1 + w J w 0 + f 0

196 190 6 Haar J {c J k } c J k J c j k j c j 1 k j 1 c 0 0 d j k j d j 1 k j 1 d 0 0 (6.21) {c j k } {cj 1 k } {d j 1 k } 1 wavelet 1-step wavelet transformation # {c j k } = # {c j 1 k } + # {d j 1 k } 1 wavelet {c j k } {cj 1 k } 2 J j(j 1 j 0) {d J 1 k J 1 },..., {d j k j }, {c j k j } 0 k j 2 j 1 j J 1 # {c J k J } = # {c J 1 k J 1 } + # {d s k s } + # {c j k j } = 2 J 1 + = 2 J. s=j ( J 1 ) 2 s + 2 j s=j j(j 1 j 0) Haar {d J 1 k J 1 },..., {d j k j }, {c j k j } w j = f j = 2 j 1 k j=0 2 j 1 k j=0 d j k j ψ(2 j x k j ), c j k j ϕ(2 j x k j ) f J f j = w j w j + f j

197 6.4 Wavelet f J {c J k } k f J w J w 0 + f 0 dyadic c 0 0 f 0 {d j l } 0 j<j 1,l=0,...,2 j 1 {w j} inverse wavelet transformation f J f f J Haar {c 0 k } k, {d j l } 0 l<j,k {c0 k } k {d j k } 0 j<j,k f J 6.45 j Haar j 1 ϕ(x) = ϕ(2x) + ϕ(2x 1) ψ(x) = ϕ(2x) ϕ(2x 1) 6.48 ϕ(2 j 1 x) = ψ(2 j x) + ϕ(2 j x 1) (6.22) ψ(2 j 1 x) = ϕ(2 j x) ϕ(2 j x 1) (6.23) 6.49 (Haar ) J f J f J = w J 1 + w J w 0 + f 0 w j (x) = k d j k ψ(2j x k) W j, 0 j < J f 0 (x) = k c 0 kϕ(x k) f J f J (x) = c J l ϕ(2 J x l) V J l Z

198 192 6 Haar c j l j = 1 j = J c j 1 c j k l = c j 1 k + d j 1 k, l = 2k, d j 1 k, l = 2k + 1. (6.24) 6.48 f 0 (x) = k = k = l c 0 kϕ(x k) c 0 k ( ) ϕ(2x 2k) + ϕ(2x 2k 1) c 1 lϕ(2x l) c 1 l = c 0 k, l = 2k c 0 k, l = 2k + 1. w 0 (x) = k = k = l d 0 kψ(x k) d 0 k ( ) ϕ(2x 2k) ϕ(2x 2k 1) d 1 lϕ(2x l) d 1 l = d 0 k, l = 2k d 0 k, l = 2k + 1. f 0 (x) + w 0 (x) = l c 1 lϕ(2x l)

199 6.4 Wavelet 193 c 0 k + d0 k, l = 2k c 1 l = c 1 l + d 1 l c 0 k d0 k, l = 2k + 1. w 1 = k d 1 l ψ(2x k) f 2 (x = f 0 (x) + w 0 (x) + w 1 (x) = lc 2 lϕ(2 2 x l) c 2 c 2 l = c 2 l + d k + d2 k, 2 l c 2 k d2 k, l = 2k l = 2k + 1. f J = f J 1 + w J f(x) = sin(2x) cos ( x 3) + Å ã 2 sin(110(x 2)) 110(x 2) Å ã 2 sin(95(x 1)) (6.25) 95(x 1) [0, π] 6.3 x = 1 x = 2 [0, π) N = 2 8 = 254 J = 8 dyadic {I 8,k } k x = k 2 8, (k = 0,..., 2 8 1) {c 8 k = f ( ) k 2 8 }k=0,...,28 1 Mathematica j ϕ(2 j x k) HaarScale[x, j, k] [0, π] f(x) (6.16) [0, 1] f(x) spiked[x] {c 8 k } coeff J = 8 f 8 (x) Mathematica 1 coeff[k + 1] HaarScale[x_Real, j_integer, k_integer] := If[x < k/2^j (k + 1)/2^j < x, 0, 1] j=8;

200 194 6 Haar (a) (b) 6.3 (a) f(x) = sin(2x) cos ( x 3) Ä ä 2 Ä ä 2 + sin(110(x 2)) 110(x 2) sin(95(x 1)) 95(x 1) [0, π] x = 1 x = 2 (b) [0, π) 2 8 {f ( kπ 2 8 ) }k=0,...,2 8 1 [0, 1) f(x) f(πx) 2 8 f 8(x) coeff = Table[N[spiked[t]], {t, 0, Pi, Pi/(2^j - 1)}]; spiked[x_real] := Sum[coeff[[k + 1]] HaarScale[x, j, k], {k, 0, 2^j - 1}] Plot[dspiky[x], {x, 0, 1}] J = 8 {c 8 k } oddcoeff evencoeff J 1 = 7

201 6.4 Wavelet 195 (7a) (7b) 6.4 (7a) (6.25) J = 8 f 8 f 7(x) (7b) w 7(x). w 7(x) f 7 (x) averaged[x] w 7 (x) detailed[x] Mathematica 1 0 even odd even - odd Mathematica odd - even [[...]] i;;j;;2 i j 2 Mathematica Haarwavelet[x_Real, j_integer, k_integer] := HaarScale[x, j + 1, 2 k] - HaarScale[x, j + 1, 2 k + 1] oddcoeff = coeff[[1 ;; Length[coeff] ;; 2]]; evencoeff = coeff[[2 ;; Length[coeff] ;; 2]]; avecoeff = (evencoeff + oddcoeff)/2; diffcoeff = (oddcoeff - evencoeff)/2; detailed7[x_real] := Sum[diffCoeff[[k + 1]] Haarwavelet[x, 7, k], {k, 0, 2^7-1}] averaged7[x_real] := Sum[aveCoeff[[k + 1]] HaarScale[x, 7, k], {k, 0, 2^7-1}]

202 196 6 Haar (6a) (6b) (5a) (5b) (4a) (4b) (3a) (3b) 6.5 (6.25) J = 8 f 8 6 f 6 w 6, 5 f 5 w 5, 4 f 4 w 4, 3 f 3 w 3.

203 6.4 Wavelet 197 (2a) (2b) (1a) (1b) (0a) (0b) 6.6 (6.25) J = 8 f 8 2 f 2 w 2, 1 f 1 w 1, 0 f 0 w 0. j = 7 detailed7[x] + averaged7[x] dspiked[x] 6.16(b) 6.4 j = {f l } dyadic I 8,k f 8 f 7 dyadic I 7,k = I 8,2k I 7,2k+1 c 8 2k c8 2k (c8 2k + c8 2k+1 ) f (c7 2k + c7 2k +1 ) f 5, f 4, f 3, f 2, f 1 f 0 [0, 1)

204 198 6 Haar (6-0a) 2 l l f l f f 0 f 6.6(0) f 0 + w 0 (1) f 1 (2) f 1 + w 1 (2) f 2 f l+1 = f l + w l (0 l < j) J f J Haar f J = w J w J k + f J k f J k (x) =averagedhaarscaling[x, k] w J k (x) =HaarWaveletElement[x, k] Mathematica J {c J k } = coeff }, {d J k l } f J k, w J k J k Haar {c J k l 1 k J k 0 j J k {c j l } {d j l } averagedhaarscaling[x_real, k_integer] := Module[{cf, n, oddc, evenc, avecf, j, i}, (* coef=table[n[f[x]], {x, a, b,(b-a)/(2^n-1)}]; *) cf = coeff; n = Log[2, Length[coeff]]; For[j = 0, j < k, j++, oddc = cf[[1 ;; 2^(n - j) ;; 2]]; evenc = cf[[2 ;; 2^(n - j) ;; 2]]; avecf = (oddc + evenc)/2; cf = avecf; ]; Sum[cf[[i + 1]] HaarScale[x, n - k, i], {i, 0, 2^(n - k) - 1}] ] HaarWaveletElement[x_Real, k_integer] := Module[{cf, n, oddc, evenc, avecf, diffcf, j, i}, (* coef=table[n[f[x]], {x, a, b,(b-a)/(2^n-1)}]; *) cf = coeff; n = Log[2, Length[coeff]]; For[j = 0, j < k, j++,

205 oddc = cf[[1 ;; 2^(n - j) ;; 2]]; evenc = cf[[2 ;; 2^(n - j) ;; 2]]; avecf = (oddc + evenc)/2; cf = avecf; ]; diffcf = (oddc - evenc)/2; Sum[diffcf[[i + 1]] Haarwavelet[x, n - k, i], {i, 0, 2^(n - k) - 1}] ] 6.5(3) Plot[averagedHaarScaling[x, 5], {x, 0, 1}] Plot[HaarWaveletElement[x, 5], {x, 0, 1} HaarWaveletElement[x, 4] HaarWaveletElement[x, 5] + averagedhaarscaling[x, 5] 6.5 {c j k } {dj k } wavelet 6.47 f j f j (x) = w j 1 (x) + f j 1 (x) Haar {f j } (6.19) (6.20) c j k = cj+1 2k + cj+1 2 d j k = cj+1 2k cj+1 2 2k+1 2k+1, (6.19). (6.20) cj+1 k j+1 c j k j d j+1 k j+1 d j J k j 2 J {c J k } (6.19) (6.20) {cj 1 k } {c J 2 k }... (6.21) c 0 0 d 0 0 Wavelet 1 wavelet 190 J j(j 1 j 0) Haar

206 200 6 Haar {d J 1 k J 1 },..., {d j k j }, {c j k j } 2 J Haar J {c J k } k=0,...,2 J 1 2 J data 1 wavelet j(0 j J 1) J = 3 0 [ j=3 c 3 0 c 3 1 c 3 2 c 3 3 c 3 4 c 3 5 c 3 6 c7] 3 j=2 j=1 j=0 î ó c 2 0 d 2 0 c 2 1 d 2 1 c 2 2 d 2 2 c 2 3 d 2 3 î c 1 0 d 2 0 d 1 0 d 2 1 î c 0 0 d 2 0 d 1 0 d 2 1 c 1 1 d 2 2 d 1 1 d 2 3 d 0 0 d 2 2 d 1 1 d wavelet j 1 data j data k tmp data[2 J j+1 k] c j 1 k data[2 J j+1 k] 1 ( tmp + data[2 J j+1 k + 2 J j ] ) 2 d j 1 k data[2 J j+1 k + 2 J j ] 1 ( tmp data[2 J j+1 k + 2 J j ] ) 2 k = 0,..., 2 j 1 1 j = 0 Haar 6.51 ( ) J 2 J {c J k } k=0,...,2 J 1 2 J data data {c J k } j = 0 Haar for j = J to 1 for k = 0 to 2^(j-1) - 1 tmp <- data[2^(j-j+1)k] ó ó

207 data[2^(j-j+1)k] <- (tmp + data[2^(j-j+1)k + 2^(J-j)]) / 2 data[2^(j-j+1)k + 2^(J-j)] <- (tmp - data[2^(j-j+1)k + 2^(J-j)]) / 2 2 J data j(0 j J 1) {d J 1 k J 1 },..., {d j k j }, {c j k j } {c j k j } = {data[2 J j k j ]}, k j = 0,... 2 j 1 {d j k j } = {data[2 J j k j + 2 J j 1 ]} J {c J k } 2J data[ ] 1 tmp 2 j + 1 data[2 (J j+1)k ] data[2 (J j+1)k + 2 (J j) ] 2 ( J 1 ) = 2 J Mathematica 1 Haar 6.49 (6.24) j(0 j J 1) Haar {d J 1 k J 1 },..., {d j k j }, {c j k j } c j 2k = cj 1 k + d j 1 k, k = 0,..., 2 j 1 1 (6.26) c j 2k+1 = cj 1 k d j 1 k (6.27) 2 J {c J k } k=0,...,2 J 1 j Haar Haar {c J k } 6.54 ( ) j Haar Haar 2 J data data {d J 1 k J 1 },..., {d j k j }, {c j k j } J {c J k } for j = 0 to J-1 for k = 0 to 2^j - 1 tmp <- data[2^(j-j)k]

208 202 6 Haar data[2^(j-j)k] <- tmp + data[2^(j-j)k + 2^(J-j-1)] data[2^(j-j)k + 2^(J-j-1)] <- tmp - data[2^(j-j)k + 2^(J-j-1)] J f J f J = w J w J k + f J k Haar Mathematica J {c J k } 1 wavelet 2j j 1 j J {c j k } 2j 1 j 1 {c j 1 k } {d J 1 k } wavelet wavelet 2 J { {d J 1 k },..., {d 1 k }, d0 0, c0} 0 {c j k } {cj 1 k } {d J 1 k } {c j k } 2j 1 wavelet 2j 1 2 {c j 1 k } 2 j 1 2 {d J 1 k } j 1 wavelet 2 J ß {c j k j } 0 kj 2 1, {d j j k j } 0 kj 2 1, {d j 1 j k j 1 } 0 kj 1 2 1,..., {d J 1 j 1 k J 1 } 0 kj 1 2 J

209 203 7 Haar 7.1 Haar 6.50 J = 8 2 J {c J k } f J Haar f J = w J w J k + f J k w J k (x) Haar {c j k } j=0,...,j 1, {d j k } j=0,...,j 1 dyadic J 4,4100Hz CD /44100 = 11.9[sec] 512 = = Fourier Haar [a, b) f(x) J [a, b) 2 J x = a + k b a (k = 1,..., 2 J 1) 2 J 2 J ã { c J k } k=0,...,2 J 1, Å cj k = f a + k b a 2 J c J 2 J 1 x = b b q 2 J f Ä b b q 2 J ä

210 204 7 Haar [a, b) f(x) 2 J {c J k } 2 J [0, 1) f J (x) = 2 J 1 k=0 c J k ϕ(2 J x k) (6.16) 6.3(b) [0, π) {c J k } [0, 1) [a, b) f [0, 1) f f [0, 1) f J [0, 1) f 2 J f = (c J 0,..., c J 2 J 1 ) [0, 1) f discretaization Haar [0, 1) 2 J {c J k } 0 j J 2 {c j k } Haar 6.47 (6.19) Haar J (6.20) {d j k } Haar 6.49 (6.24) {c J k } [0, 1) Haar 2 J Haar ( Haar ) J 2 J+1 (1) Haar ϕ J (j, k) ϕ J (j, k) = ( 0 }. {{.. 0 }} 1.. {{ } 0 }.. {{ } ), (0 j J, 0 k < 2 j 1) k2 J+1 j 2 J+1 j 2 J+1 (k+1)2 J+1 j (7.1) (2) Haarwavelet ψ J (j, k)

211 7.1 Haar 205 ψ J (j, k) = ( 0 }. {{.. 0 } 1 }. {{.. 1 }, }{{}}.. {{ } ), (0 j J, 0 k < 2 j 1) k2 J+1 j 2 J j 2 J j 2 J+1 (k+1)2 J+1 j (7.2) 7.2 ϕ J (0, 0) ψ J (0, 0) [0, 1) ϕ(x) ψ(x) ψ(2 j x) ϕ(2 j x) ψ(2 j x) 1 2 j ±1 J Haar 2 J+1 j Haar ϕ J (j, k) ψ J (j, k) 2 J+1 x = k, k = 0,..., 2 J J (J+1) 0 j J Haar ϕ(2 j x k) ψ(2 j x k) J 2 J+1 Haar (1.1) Haar J Haar (1) V J j = span{ϕ J (j, k)} k=1,...,2j 1 ϕ J (j, k) ϕ J (j, k ) = 2 J+1 j δ kk, (0 j < J, 0 k, k < 2 j 1). (2) {Wj J} j=0,...,j = { } span{ψ J (j, k)} k=1,...,2j 1 j=0,...,j W J j W J j ψ J (j, k) ψ J (j, k ) = 2 J+1 j δ jj δ kk, (0 j, j < J, 0 k < 2 j 1, 0 k < 2 j 1). (3) {ϕ J (j, k)} k=1,...,2 j 1, ψ J (j, k )} 0 j j <J, k=0,...,2 j 1, k =0,...,2 j 1 Vj J Wj J (j j ). ϕ J (j, k) ψ J (j, k ) = 0, (j j, 0 k < 2 j 1, 0 k < 2 j 1). Haar

212 206 7 Haar (a) (b) (c) (d) 7.1 (a) J = 3 Harr ϕ 3(2, 2), (b) ϕ 3(3, 7). (c) Harr wavelet ψ 3(2, 2), (d) ψ 3(7, 3). J = 3 2 J+1 = 16 k2 J+1 j + 1 (k + 1)2 J+1 j k = 0,..., 2 j 1 2 J ϕ J(J, k) wavelet ψ J(J, k) ϕ J(j, k) ψ J(j, k) x = k, k = 0,..., 2 J J+1

213 7.1 Haar Haar Haar Haar 2 J {c J k } k=0,...,2 J 1 {d J 1 k J 1 },..., {d j k j }, {c j k j }, (0 j J 1) [0, 1) J 1 2 j 1 j =j k j =0 2 j 1 d j k j ψ(2j x k j ) + 2 J+1 J 1 2 j 1 j =j k j =0 k=0 2 d j k ψ j 1 j J(j, k j ) + k=0 c j k ϕ(2j x k) c j k ϕ J(j, k) Haar [0, 1) ϕ(2 j x k) ϕ J (j, k) ψ(2 j x k) ψ J (j, k) J {c J k } k=0,...,2 J 1 {d J 1 k J 1 },..., {d j k j }, {c j k j }, (0 j J 1) Haar Mathematica J = scale Haar ϕ scale (j, k) = dhaarscale[j, k, scale] ψ scale (j, k) = dhaarwavelet[j, k, scale] dhaarscale[j_integer, k_integer, scale_integer] := Module[{}, hs[j] := Module[{bhalf, ehalf, i}, bhalf = Table[1, {i, 1, 2^(scale j)}]; ehalf = Table[0, {i, 2^(scale j) + 1, 2^(scale + 1)}]; Flatten[Append[bhalf, ehalf]] ]; Nest[RotateRight, hs[j], 2^(scale j)*k] ] dhaarwavelet[j_integer, k_integer, scale_integer] := Module[{}, hw[j] := Module[{bquater, equater, remain, i}, bquater = Table[1, {i, 1, 2^(scale - j)}]; equater = Table[-1, {i, 2^(scale - j) + 1, 2^(scale j)}]; remain = Table[0, {i, 2^(scale j) + 1, 2^(scale + 1)}];

214 208 7 Haar Flatten[{bquater, equater, remain}] ]; Nest[RotateRight, hw[j], 2^(scale j)*k] ] ϕ 3 (2, 2) ψ 3 (2, 2) dhaarscale[2, 2, 3] {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0} dhaarwavelet[2, 2, 3] {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 0, 0, 0, 0} ListPlot[dHaarScale[2, 2, 3], Filling -> Axis] dhaarwavelet[2, 2, 3] x = 1, 2,..., 16 Mathmatica x = 1 J 2 J x = k, k = 2 J+1 0,..., 2 J+1 1 ListPlot plotrightposition[lst] plotrightposition[lst_list] := Table[{(i - 1)/Length[lst], lst[[i]]}, {i, 1, Length[lst]}] ListPlot[plotRightPosition[dHaarScale[2, 2, 3]], Filling -> Axis] 2 (J+1) J ϕ 3 (2, 2) ψ 3 (2, 2) Haar 2 V W tensor product V W V W (7.3)

215 7.2 2 Haar 209 V W v V w W v w v V w W (av 1 + bv 2 ) w = a(v 1 w) + b(v 1 w), v (cw 1 + dw 2 ) = c(v w 1 ) + d(v w 2 ) 7.7 ( ) 1 2 V W f V g W (f g)(x, y) = f(x)g(y) (7.4) 7.8 ( (af 1 + bf 2 ) g ) (x, y) = a(f 1 g)(x, y) + b(f 2 g)(x, y) (x, y) (f g)(ax 1 + bx 2, y) af(x 1 )g(y) + bf(x 2 )g(y). 7.9 ( Kronecker ) m V n W u = [u 0,..., u m 1 ] V, v = [v 0,..., v n 1 ] W Kronecker u v 0 u 0 v 0... u 1 v k... u 0 v n 1 u v = u j v 0... u j v k... u j v n 1 u m 1 v 0... u m 1 v k... u m 1 v n 1. (7.5) 7.10 (7.5) u v m n 7.11 m n M A = (a i,j ), B = (b i,j ) M 0 A B = m 1 n 1 a i,jb i,j (7.6) i=0 j=0 A B

216 210 7 Haar 7.12 a, b, c, d a b c d a b c d = (a b) ij (c d) ij = (a i b j )(c i d j ) i j i j ( ) ( ) = a i c i b j d j = a c b d i j 7.13 (7.1) J Haar ϕ J (J, k) ψ J (J, k) Kronecker 0 k 2 J 1 ϕ J (J, k), ψ J (J, k) J 2 J Haar 7.1 J = 0 ï ò ï ò ϕ 0 (0, 0) ϕ 0 (0, 0) =, ϕ (0, 0) ψ 0 (0, 0) =, 1 1 ï ò ï ò ψ 0 (0, 0) ϕ 0 (0, 0) =, ψ (0, 0) ψ 0 (0, 0) = mat Mathematica ListPlot3D[mat, Mesh -> None, InterpolationOrder -> 1, ColorFunction -> "SouthwestColors"] 7.2 (a ), (b ), (c ), (d ) Haar Haar Kronecker Haar ϕ(x), ψ(x) J = 0 Haar ϕ 0 (0, 0), ψ 0 (0, 0) 7.2

217 7.2 2 Haar 211 (a) (a ) (b) (b ) (c) (c ) (d) (d ) Haar 2 Haar (a) ϕ ϕ(x, y) (a ) ϕ 0(0, 0) ϕ ) (0, 0), (b) ϕ ψ(x, y) (b ) ϕ 0(0, 0) ψ ) (0, 0), (c) ϕ ψ(x, y) (c ) ψ 0(0, 0) ϕ ) (0, 0), (d) ψ ψ(x, y) (d ) ψ 0(0, 0) ψ ) (0, 0). 7.2 Haar Haar

218 212 7 Haar J = 1 k, l = 0, 1 ϕ 1 (1, k) ϕ 1 (1, l) ϕ 1 (1, 0) ϕ 1 (1, 0) = , ϕ 1(1, 0) ϕ 1 (1, 1) = , ϕ 1 (1, 1) ϕ 1 (1, 0) = , ϕ 1(1, 1) ϕ 1 (1, 1) = ϕ 1 (1, k) ψ 1 (1, l) ϕ 1 (1, 0) ψ 1 (1, 0) = , ϕ 1(1, 0) ψ 1 (1, 1) = , ϕ 1 (1, 1) ψ 1 (1, 0) = , ϕ 1(1, 1) ψ 1 (1, 1) = ψ 1 (1, k) ϕ 1 (1, l) ψ 1 (1, 0) ϕ 1 (1, 0) = , ψ 1(1, 0) ϕ 1 (1, 1) = , ψ 1 (1, 1) ϕ 1 (1, 0) = , ψ 1(1, 1) ϕ 1 (1, 1) = ψ 1 (1, k) ψ 1 (1, l) ψ 1 (1, 0) ψ 1 (1, 0) = , ψ 1(1, 0) ψ 1 (1, 1) = , ψ 1 (1, 1) ψ 1 (1, 0) = , ψ 1(1, 1) ψ 1 (1, 1) =

219 7.2 2 Haar 213 J 2 J+1 Haar ϕ J (J, k) ψ J (J, l) Φ J k,l ϕ J (J, k) ϕ J (J, l) = ( }{{} 2k }{{} 2 J+1 2(k+1) ) (0 }. {{.. 0 } 2l }.. {{ } ) (7.7) 2 J+1 2(l+1) 2 J+1 2 J+1 0 k, l 2 J (J+1) 0 ( ) Φ J k,l = m,n 1, 2k m 2k + 1, 2l n 2l + 1, 0,. (7.8) ϕ J (J, k) ϕ J (J, l) 2 2 Ψ J,d k,l ϕ J(J, k) ψ J (J, l), (7.9) Ψ J,v k,l ψ J(J, k) ϕ J (J, l), (7.10) Ψ J,d k,l ψ J(J, k) ψ J (J, l) (7.11) 2k m 2k+1, 2l n 2l+1, 0 k, l 2 J (2k, 2l)- ( ) ï ò Φ J 1 1 k,l = 1 1 ( ) ï ò Ψ J,h 1 1 k,l = 1 1 ( ) ï ò Ψ J,v 1 1 k,l = 1 1 ( ) ï ò Ψ J,d 1 1 k,l = Φ J k,l Φ J k,l ΦJ k,l = Ψ J,h k,l = ΨJ,v k,l = ΨJ,d k,l (7.12) (7.13) (7.14) (7.15) = 2 2 = 4 (7.16)

220 214 7 Haar Haar (7.1) ϕ J (j, k) = ψ J (j, l) = 2 J+1 j Haar 2 J+2 M = (m i,j ) 2 J+2 2 J+2 = 2 2(J+2) ( E(i, j) ) k,l = 1, k = i, l = j 0, M = 2 2(J+2) i,j=1 m i,j E(i, j) Haar 7.16 Haar 2 J+1 (J 0) { Φ J k,l = ϕ J (J, k) ϕ J (J, l), Ψ J,h k,l = ϕ J(J, k) ψ J (J, l), Ψ J,v k,l = ψ J(J, k) ϕ J (J, l), Ψ J,d k,l = ψ J(J, k) ψ J (J, l) } 0 k,l 2 J 1 2 J+1 2 2(J+1) 2 J+1 2 2(J+1) 2 2(J+1) 2 2J {ϕ J (j, k)} {ψ J (j, k)} ΦJ k,l Ψ J,h k,l = ϕ J (J, k) ϕ J (J, l) ϕ J (J, k ) ψ J (J, l ) = ϕ J (J, k) ϕ J (J, k ) ϕ J (J, l) ψ J (J, l ) = 2 2 δ kk δ ll.

221 7.3 2 Haar J+1 (J 0) A A = 2 J 1 k,l=0 + 1 ( Φ J 4 k,l A A Ψ J,d k,l ΦJ k,l + ) Φ J,d k,l Ψ J,h k,l A Φ J,h J,v k,l + Ψk,l A Φ J,v k,l (7.17) 1 2 ΦJ k,l, 1 2 ΨJ,h k,l, 1 2 ΨJ,v k,l 1 2 ΨJ,d k,l J+1 A = (a i,j ) Φ J k,l, ΨJ,h k,l, ΨJ,v k,l ΨJ,d k,l 0 k, l 2J 1 ( ) ΦJ k,l A a2k,2l + a 2k,2l+1 + a 2k+1,2l + a 2k+1,2l ΨJ,h 4 k,l A ( ) = 1 4 a2k,2l a 2k,2l+1 + a 2k+1,2l a 2k+1,2l+1 1 ΨJ,v 4 k,l A ( ) (7.18) = 1 4 a2k,2l + a 2k,2l+1 a 2k+1,2l a 2k+1,2l+1 1 ΨJ,d 4 k,l A ( ) = 1 4 a2k,2l a 2k,2l+1 a 2k+1,2l + a 2k+1,2l J = 2 2(J+1) 2 J+1 Haar wavelet = (1 Haar wavelet ) 2 J+1 A A Haar ã ÅA Haar = 1 Å ã Φ J k,l 4 k,l A, A Haar = 1 ΨJ,h k,l A k,2 J +l 4 ã ÅA Haar = 1 ΨJ,v k,l A ã, ÅA Haar = 1 ΨJ,d k,l A. 2 J +k,l 4 2 J +k,2 J +l 4 0 k, l 2 J 1 A Haar A 1 Haar wavelet 1 Haar wavelet H : A A Haar (7.19)

222 216 7 Haar 7.19 J = 0 A = [ a c ] b (7.18) d 1 4 A Haar = (a + b + c + d) 1 4 (a b + c d) 1 4 (a + b c d) 1 4 (a b c + d) A 1 wavlelet wavlelet wavelet 1 2 (a + b) 1 2 (a b) 1 2 (c + d) 1 4 (c d) 1 wavelet Haar ( (a + b) (c + d)) 1 2 ( 1 2 (a b) (c d)) ( (a + b) 1 2 (c + d)) 1 2 ( 1 2 (a b) 1 2 (c d)) 7.20 J = A = A Haar = A 1 wavlelet 1 wavlelet 4 4. Haar.

223 7.3 2 Haar wavelet 1 (9 + 7) 1 (6 + 2) 1 (9 7) 1 (6 2) (5 + 3) 1 (4 + 4) 1 (5 3) 1 (4 4) (8 + 2) 1 (4 + 0) 1 (8 2) 1 (4 0) (6 + 0) 1 (2 + 2) 1 (6 0) 1 (2 2) wavelet ( ) (9 + 7) + (5 + 3) 1 4 ( ) 1 4 (8 + 2) + (6 + 0) ( ) 1 4 (9 + 7) (5 + 3) ( ) (8 + 2) (6 + 0) 1 4 ( ) 1 4 (6 + 2) + (4 + 4) ( ) (4 + 0) + (2 + 2) ( (6 + 2) (4 + 4) ) ( (4 + 0) (2 + 2) ) ( ) 1 4 (9 7) + (5 3) ( ) (8 2) + (6 0) ( (9 7) (5 3) ) ( (8 2) (6 0) ) ( ) (6 2) + (4 4) ( ) (4 0) + (2 2) ( ) (6 2) (4 4) ( ) (4 0) (2 2) [ ] 6 4 A Haar 2 2 Haar wavelet 4 2 [ ] wavelet A HA = A Haar H 2 A = A Haar 2 = H H Haar j+1 P j 1 Haar wavelet HP j

224 218 7 Haar 2 j 4 0 k, l 2 j 1 ( P j 1 ) k,l = 1 Φ J 4 k,l P j, ( Q j 1,h ) k,l = 1 4 ( Q j 1,v ) k,l = 1 ΨJ,v k,l P j (, Q j 1,d ) 4 k,l = 1 4 P j H HP j = P j 1 Q j 1,v Q j 1,h Q j 1,d Haar. ΨJ,h k,l ΨJ,d k,l P j, P j 2 J+1 P J Haar wavelet J Haar wavelet HP j = P j 1 {P J,..., P 1 } Haar Haar Haar 2 N 1 Wavelet N/2 4 LL 1, LH 1,HL 1 HH 1 LL 1 1 Haar Haar2DWaveletTransform[signal_List] := Module[{datasize, scale, hdata, vdata, m, n}, datasize = Length[signal]; (*scale=log[2,datasize];*) hdata = ConstantArray[0, {datasize, datasize}]; For[m = 0, m <= datasize - 1, m++,(* Wavelet *) For[n = 0, n <= datasize/2-1, n++, hdata[[m + 1, n + 1]] = (signal[[m + 1, 2 n + 1]] + signal[[m + 1, 2 n + 2]]) / 2; hdata[[m + 1,datasize/2 + n + 1]] =

225 7.4 2 Haar 219 ] (signal[[m + 1, 2 n + 1]] - signal[[m + 1, 2 n + 2]]) / 2; ] ]; vdata = hdata; For[n = 0, n <= datasize/2-1, n++,(* Wavelet *) For[m = 0, m <= datasize/2-1, m++, vdata[[m + 1, n + 1]] = (hdata[[2 m + 1, n + 1]] + hdata[[2 m + 2, n + 1]]) / 2; vdata[[datasize/2 + m + 1,n + 1]] = (hdata[[2 m + 1, n + 1]] - hdata[[2 m + 2, n + 1]]) / 2; vdata[[m + 1,datasize/2 + n + 1]] = (hdata[[2 m + 1, datasize/2 + n + 1]] + hdata[[2 m + 2, datasize/2 + n + 1]]) / 2; vdata[[datasize/2 + m + 1, datasize/2 + n + 1]] = (hdata[[2 m + 1, datasize/2 + n + 1]] - [[2 m + 2, datasize/2 + n + 1]]) / 2; ; ]; vdata 5.4 Mathematica data data/construction.jpg 2 9 = 512 graydata Image[graydata] graydata 1 Haar SetDirectory[NotebookDirectory[]] const = Import["data/construction.jpg"] grayconstruction = ColorConvert[const, "Grayscale"] graydata = Take[ImageData[grayconstruction], {1, 512}, {1, 512}] Image[graydata] tgray = Haar2DWaveletTransform[graydata]; ArrayPlot[tgray]

226 220 7 Haar 7.3 (1) 1 Haar (2) LL 1 (a) (b) 7.3 (1) 2 9 = 512 (2) 1 Haar LL 1 LL 1 1 Haar LL 2 1 Haar 1 1 Haar N 2 Haar HaarWavelet2DTransform[signal] signal N 2 HaarWavelet2DTransform[signal_List] := Module[{datasize, maxlevel, data, tdata, j}, datasize = Length[signal]; maxlevel = Log[2, datasize]; data = signal; data = Haar2DWaveletTransform[data]; For[j = 1, j < maxlevel, j++,

227 7.4 2 Haar 221 ]; data ] tdata = Take[data, {1, datasize/2^j}, {1, datasize/2^j}]; data[[1 ;; datasize/2^j, 1 ;; datasize/2^j]] = Haar2DWaveletTransform[tdata]; 2 Haar 1 Haar Haar2DInverseWaveletTransform[vdata] Haar2DInverseWaveletTransform[vdata_List] := Module[{datasize, maxlevel, signal, m, n}, datasize = Length[vdata]; (*maxlevel=log[2,datasize];*) signal = ConstantArray[0, {datasize, datasize}]; For[n = 0, n <= datasize/2-1, n++, For[m = 0, m <= datasize/2-1, m++, signal[[2 m + 1, 2 n + 1]] = vdata[[m + 1, n + 1]] + vdata[[datasize/2 + m + 1, n + 1]] + vdata[[m + 1, datasize/2 + n + 1]] + vdata[[datasize/2 + m + 1, datasize/2 + n + 1]]; signal[[2 m + 2, 2 n + 1]] = vdata[[m + 1, n + 1]] - vdata[[datasize/2 + m + 1, n + 1]] + vdata[[m + 1, datasize/2 + n + 1]] - vdata[[datasize/2 + m + 1, datasize/2 + n + 1]]; signal[[2 m + 1, 2 n + 2]] = vdata[[m + 1, n + 1]] + vdata[[datasize/2 + m + 1, n + 1]] - vdata[[m + 1, datasize/2 + n + 1]] - vdata[[datasize/2 + m + 1, datasize/2 + n + 1]]; signal[[2 m + 2, 2 n + 2]] = vdata[[m + 1, n + 1]] - vdata[[datasize/2 + m + 1, n + 1]] - vdata[[m + 1, datasize/2 + n + 1]] + vdata[[datasize/2 + m + 1, datasize/2 + n + 1]]; ]

228 222 7 Haar ] ]; signal 2 Haar 1 Haar2DInverseWaveletTransform 2 Haar HaarInverseWavelet2DTransform[vdata] HaarInverseWavelet2DTransform[vdata_List] := Module[{size, scale, wdata, idata, j}, size = Length[vdata]; scale = Log[2, size]; wdata = vdata; For[j = 1, j <= scale, j++, idata = Take[wdata, {1, 2^j}, {1, 2^j}]; idata = Haar2DInverseWaveletTransform[idata]; wdata[[1 ;; 2^j, 1 ;; 2^j]] = idata; ]; wdata ] 5.4 Fourier approxhaardatabyneglectingsmallelements[graydata,thresholdratio] 2 data 1 thresholdratio 2 Haar maxabs maxabs thresholdratio Ignoring rate Ignoring rate approxhaardatabyneglectingsmallelements[data_,thresholdratio_] := Module[{tgraydata, replaced, ignorgraydata}, ModifiedDataNeglectingRelativeSmallElement[lst_List] := Module[{maxabs, approxedlist, replacesymbol}, maxabs = Max[Abs[lst]];

229 7.4 2 Haar 223 ] approxedlist = lst /. x_ /; Abs[x] < thresholdratio*maxabs -> replacesymbol; replaced = Count[Flatten[approxedlist], replacesymbol]; approxedlist /. replacesymbol -> 0 ]; tgraydata = HaarWavelet2DTransform[data]; ignorgraydata = ModifiedDataNeglectingRelativeSmallElement[tgraydata]; Print["Ignoring rate=", replaced / Length[Flatten[graydata]] // N]; Image[HaarInverseWavelet2DTransform[ignorgraydata]] 2 graydata 5% Haar approxhaardatabyneglectingsmallelements[graydata, 0.05] Ignoring rate= (1) (4) (1) 0.05 (2) 0.10 (3) 0.15 (4) 0.20 Haar

230 224 7 Haar (1) (2) (3) (4) Haar Haar (1) 0.05 Ignoring rate= (2) 0.10 Ignoring rate= (3) 0.15 Ignoring rate= (4) 0.20 Ignoring rate=

231 Haar 6.7 Haar ϕ j,k (x) = 2 j/2 ϕ(2 j x k) (6.3 ) Haar wavelet 6.18 Haar wavelet ψ j,k (x) = 2 j/2 ψ(2 j x k) (6.5 ) ϕ(x) ψ(x) γ(x) L 2 {T k γ(x) = γ(x k)} k {T k γ(x) = γ(x k)} k γ(x) L 2 {T k γ(x) = γ(x k)} k Z γ(x) γ(x) f(x) span{t k γ(x)} f(x) = f T k γ T k γ(x) k Z f(x) 1 γ(x)

232 {T k γ(x)} γ(x) L 2 γ(λ + 2πk) 2 = 1, λ R. (8.1) 2π k Z γ(λ) γ(x) Fourier {T k γ(x)} T k γ T l γ = γ (x k)γ(x l) dx = δ kl. Plancherel 3.22 f(x), g(x) L 2 dx f (x)g(x) = dλ f (λ)ĝ(λ). (3.13) Fourier ÿ f(x n) = e inλ f(x) δ n0 = = k Z dλ [ e inλ γ(λ) ] γ(λ) = dλ e inλ γ(λ) 2 2πk+π 2πk π E(λ) 2π k Z dλ e inλ γ(λ) 2 = γ(λ + 2πk) 2 π π dλ k Z γ(λ + 2πk) 2 (8.2) (8.2) 1 π 2π π E(λ)e inλ dλ = δ 0n (8.3) E(λ) 2π E(λ) Fourier (2.4) 1/ 2π n ε n e inλ ε n = 1/ 2π π π E(λ)e inλ dλ (8.3) ε n = δ 0n

233 8.2 MRA) 227 E(λ) = 1/ 2π E(λ) E(λ + 2π) = 2π γ(λ + 2π(k + 1)) 2 k Z = 2π γ(λ + 2πk ) 2 = E(λ). k Z 8.2 MRA) 6.1 Haar Haar wavelet 6.4 Wavelet Haar ( [19]) R L 2 {V j } j Z ϕ R (MRA: Multi Resolution Analysis) (a) j Z V j V j+1. j + 1 j (b) j Z V j = L 2 (R). ˆPj V j f(x) L 2 (R) lim j ˆPj f = f. R f(x) f(x) span{v j } j Z ε > 0 f g < ε j Z g(x) V j (c) j Z V j = {0}. (d) j Z f(x) V j f(2x) V j+1 f(x) V j f(2 j x) V 0. (e) R L 2 ϕ(x) V 0 {T k ϕ(x)} V 0 V 0 = span{t k ϕ(x)} k Z

234 228 8 J j j 8.2(b) f L 2 j V j 6 Haar 6.25 V j = { h L 2 (R) k Z h(x) x [2 j k,2 j (k+1)) = } (6.6 ) 8.2(e) ϕ(x) Haar ϕ Haar Haar [9] MRA 8.2 ϕ(x) j, k Z ϕ j,k (x) ϕ j,k (x) = 2 j/2 ϕ(2 j x k) = D 2 j T k ϕ(x), j Z (8.4) L 2 (R) f(x) j Z V j ˆP j ˆP j = ϕ j,k ϕ j,k (8.5) k Z ˆQ j ˆQ j = ˆP j+1 ˆP j (8.6) 8.4 j Z {ϕ j,k (x)} j,k Z V j ϕ 0,k (x) V 0 8.2(d) ϕ j,k (x) V j. {ϕ 0.k (x)} ϕ 0,k ϕ 0,l = ϕ j,k ϕ j,l = δ kl. {ϕ j,k (x)} j,k Z f j (x) V j D 2 j f j (x) = f j (2 j x) V j V 0 D 2 j f j = k ϕ 0,k D 2 j f j ϕ 0,k = k D 2 j ϕ 0,k f j ϕ 0,k.

235 8.2 MRA) 229 f j V j V j f j = D 2 j D 2 j f j = D 2 j D 2 j ϕ 0,k f j ϕ 0,k = D 2 j ϕ 0,k f j D 2 j ϕ 0,k k k = k ϕ j,k f j ϕ j,k = ˆP j f j MRA 8.4 ϕ 8.2 {V j } j Z {ψ} j,k Z L 2 (R) L 2 (R) f V j ˆP j ˆP j f = ˆP j 1 f + f ψ j,k ψ j,k (8.7) k Z {ψ j,k } j,k Z MRA Haar 6.25 V j V j 1 W j 1 V j = V j 1 W j 1, V j 1 W j 1. MRA 8.2(a) 6.26 (6.8) n V j = W j k V j n k=1 j J = W j k V J, j > J, (8.8) k=1 L 2 (R) = k Z W k (8.9)

236 230 8 {ψ j,k } j,k Z L 2 (R) {ψ 0,k } + k Z W 0 j Z {ψ j,k } + k Z W j 8.4 ϕ W 0 ψ W MRA 8.2(a)(d)(e) 8.5 (2- ) {V j } j Z ϕ MRA 2- ϕ(x) = p k ϕ(2x k), p k = ϕ 1,k ϕ = 2 k Z ϕ (2x k)ϕ(x) dx (8.10) ϕ(2 j 1 x l) = k Z p k 2l ϕ(2 j x k) (8.11) 8.3 (8.4) (8.11) V j ϕ j 1,l (x) = 2 1/2 k Z p k 2l ϕ j,k (x) (8.12) ϕ V 0 V 1 {ϕ 1,k } V 1 ϕ(x) ϕ(x) = k p kϕ 1,k (x). {ϕ 1mk } k Z p k = ϕ 1,k ϕ = 2 1/2 ϕ (2x k)ϕ(x) dx ϕ(x) = p k ϕ 1,k (x) k Z = ò ï2 1/2 ϕ (2x k)ϕ(x) dx 2 1/2 ϕ(2x k). k Z

237 p k = 2 1/2 p k = 2 ϕ (2x k)ϕ(x) dx x 2 j 1 x l 8.6 Haar ϕ(x) = χ [0,1)] (x) p 0 = p 1 = 1 ϕ(x) = ϕ(2x) + ϕ(2x 1). 8.7 (2- ) {V j } j Z ϕ MRA konotoki, 2- (1) p k 2lp k = 2δ l0 k Z (2) p k 2 = 2 k Z (3) k Z p k = 2 (4) k Z p 2k = 1, p 2k+1 = 1. k Z (1) 1.60 Parseval f g = k f k g k (2) l = 0 (1) (3) 8.5 ϕ(x) dx = p k ϕ(2x k) dx k Z = 1 p k ϕ(x ) dx 2 k Z ϕ(x )dx (4) k p k 2l p k = 2δ l0 l l l p k+2lp k = 2. l Z k Z

238 232 8 k 2 = ( p 2k+2lp 2k + ) p 2k+1+2lp 2k+1 l Z k Z k Z ) ) = k Z ( l Z p 2k+2l p 2k + k Z ( l Z p 2k+1+2l p 2k+1 l l k 2 = p 2k p 2l + p 2k+1 p 2l+1 k Z l Z k Z l Z 2 2 = p 2k + p 2k+1 k Z k Z E = k p 2k O = k p 2k+1 2 = E 2 + O 2 (3) E + O = 2. E, O 2 E = O = MRA J j V j+1 {V j } j Z 8.2 V j+1 V j W j MRA ϕ W j {ψ j,k } k Z {ψ j,k } k Z {V j } j Z 2- ϕ(x) = p k ϕ(2x k) (8.10 ) k Z ϕ V j+1 V j W j V j W j ψ(x) = k Z( 1) k p 1 kϕ(2x k) (8.13) {ϕ(2 j x k)} k Z ψ j,k (x) = 2 j/2 ψ(2 j x k) (8.14)

239 Haar ϕ(x) p 0 = p 1 = 1. Haar ψ(x) = p 1 ϕ(2x) p 0 ϕ(2x 1) j = 0 MRA 8.2(d) j {ψ 0,k (x) = ψ(x k)} k Z ψ 0,k (x) k Z V 0 W 0 V 1 V 0 (8.14) ψ 0,m (x) = k p ( 1)k 1 k +2m ϕ(2x k ) {2 1/2 ϕ(2x k)} k Z 8.7 (1) ψ 0,m ψ 0,l = 1 p 1 k+2m p 1 k+2l 2 k Z = 1 i Zpi p i+2l 2m 2 = δ m l,0 = δ m,l {ψ 0,k (x) = ψ(x k)} k Z ψ(x m) V 0 ϕ(x l) ψ 0,m ϕ 0,l = 1 ( 1) k p 1 k+2m p k 2l. 2 k Z m = l = 0 k( 1) k p 1 k p k j 0 k = m + l j k = m + l + j + 1 W 0 V 0 V 1 ϕ(2x j) V 0 ϕ(x k) W 0 ψ(x k) ϕ(2x j) = k a k ϕ(x k) + b k ψ(x k) V 1 = V 0 W 0 {ϕ(x

240 234 8 k)} k Z a k = ϕ (y k)ϕ(2y j) dy = p l ϕ (2y 2k l)ϕ(2y j) dy l = 1 2 p j 2k b k = ψ (y k)ϕ(2y j) dy = 1 2 ( 1)j p 1 j+2k. ϕ(2x j) = 1 p 2 j 2kϕ(x k) + ( 1) j p 1 j+2k ψ(x k) k = 1 ) (( 1) j+1 p 1 j+2k p 1 l + p 2 j 2kp l ϕ(2x 2k l). k,l ϕ(2x 2k l) 2k + l = j l = j 2k p 1 j+2k p 1 j+2k + p j 2kp j 2k = 2 (*) k n 0 l = j 2k + n ( 1) n p 1 j+2k p 1 j+2k n + p j 2kp j 2k+n = 0, k k n 0 (**) ( ) m = 1 j + 2k m = j 2k 8.7 (2) p m p m = 2 m ( ) n n = 2i k = i + j k k p j 2k +np j 2k

241 n = 2i k = k + j + i k p j+2k np j+2k m = j + 2k ( ) m = 1 j + 2k ( ) m p m 2ip m 8.7 (1) 8.8 (8.13) (8.14) W j ψ j,k (x) = 2 1/2 ψ(2 j x k) V j+1 ϕ j+1,l (x) ψ j,k (x) = 2 1/2 l Z( 1) l p 1 l+2kϕ j+1,l (x) (8.15) V j V j 1 W j 1 V j = V j 1 W j 1 j J V j = W j k V J, j > J, (8.8 ) k=1 {V j } V 1, V 2,... J < 0 MRA 8.2(c) (8.9) 8.10 {V j } j Z ϕ MRA W j V j+1 V j L 2 (R) = W k. (8.9 ) k Z j Z {ψ j,k } k Z W j {ψ j,k (X)} j,k Z L 2 (R) f L 2 (R) k Z w k, w k W k

242 MRA f j (x) j Z V j f L 2 (R) V j ˆP j f j (x) = ˆP j f(x) 8.4 f j V j f j (x) = ϕ j,k f ϕ j,k (x). (8.16) k Z V j = V j 1 W j 1, V j 1 W j f j f j = f j 1 + w j 1, f j 1 V j 1, w j 1 W j 1 V j 1 W j 1 f j (x) = f j 1 (x) + w j 1 (x) = ϕ j 1,k f ϕ j 1,k (x) + ψ j 1,k f ψ j 1,k (x). (8.17) k Z k Z (8.12) (8.14) (8.17) ϕ j 1,k f = 2 1/2 k Z p k 2l ϕ j,k f ψ j 1,k f = 2 1/2 k Z( 1) k p 1 k+2l ϕ j,k f } (8.18) f j (x) ϕ j,k (x) ϕ j 1,k ϕ j,k = 2 1/2 p k 2l ψ j 1,k ϕ j,k = 2 1/2 ( 1) k p 1 k+2l (8.18) ϕ j,k (x) = 2 1/2 p k 2lϕ j 1,k (x) + 2 1/2 ( 1) k p 1 k+2l ψ j 1,k (x) (8.19) l Z l Z (8.19) (8.12) 2- (8.16) f j (x) ϕ j,k f ϕ j,k f =2 1/2 p k 2l ϕ j 1,l f l Z + 2 1/2 l Z( 1) k p 1 k+2l ϕ j 1,l f (8.20)

243 ( ) MRA f j V j {c j k = 2j/2 ϕ j,k f } f j (x) = k Z 2 j/2 ϕ j,k f 2 1/2 ϕ j,k (x) = k Z c j k ϕ(2j x k) (8.17) {d j 1 k f j (x) = k Z = 2 (j 1)/2 ψ j 1,k f } c j 1 k ϕ(2 j 1 x k) + d j 1 k ψ(2 j 1 x k). k Z : : c j k = l Z {c j 1 k = 1 p 2 k 2la j k k Z d j 1 k = 1 ( 1) k p 1 k+2l a j k 2 k Z (8.21) p k 2l a j 1 l + ( 1) k p 1 k+2lb j 1 l (8.22) l Z MRA f L 2 (R) f 3.5 MRA V j f V j V j ˆP j V j ˆP j f(x) = k Z c j k ϕ(2j x k), c j k = 2j ϕ )2 j x k)f(x) dx (8.23) f L 2 (R) {c j k } 6.4 c j k Cf(k/2j ), C = ϕ (x) dx

244 238 8 ϕ(x) x m ϕ M c j k = ϕ (x )f(2 j x + 2 j k) dx M j x [ M, M] 2 j x + 2 j k +2 j k f(2 j x + 2 j k) f(2 j ) M c j k f(k/2j ) ϕ (x ), dx M (8.23) ˆP j f(x) f j (x) = C k Z f(k/2 j )ϕ(2 j x k) (8.23 ) (6.21) f j f f j f j 1 f j 2 f 1 f 0 w j 1 w j 2 w 1 w 0 (8.24) 8.12 s = (..., s 2, s 1, s 0, s 1, s 2,... ) ˆD ˆDs = (..., s 2, s 0, s 2, ) (8.25) 5.2(6) x = (..., x 1, x 0, x 1,...), y = (..., y 1, y 0, y 1,...) (x y) i = l Z x l y i l (8.19) h = (..., h 1, h 0, h 1,...) l = (..., l 1, l 0, l 1,...)

245 h k = 1 2 ( 1)k p k+1 (8.26) l k = 1 2 p k (8.27) c j = (..., c j 1, cj 0, cj 1,... ) dj 1 = (..., d j 1 1, dj 1 0, d j 1 1,... ) (8.21) c j 1 = D(l c j ) (8.21 ) d j 1 = D(h c j ). (8.22 ) f 0 f 1 f 2 f j 1 f j f w 0 w 1 w 2 w j 1 (8.28) h k = 1 2 ( 1)k p 1 k (8.29) l k = 1 2 p k (8.30) (8.22) c j k = l Z l k 2l c j 1 l + h k 2l d j 1 l (8.31) l Z 8.13 s = (..., s 2, s 1, s 0, s 1, s 2,... ) Û Ûs = (..., s 2, 0, s 1, 0, s 0, 0, s 1, 0s 2, 0 ) (8.32) (8.31) a j = l (Ua j 1 ) + h (Ud j 1 ) (8.33)

246

247 241 9 Daubechies 9.1 Fourier (8.10) ϕ(x) = k Z p k ϕ(2x k) (8.10 ) Fourier ϕ(x) = k Z p k ϕ(2x k) Fourier H(z) = 1 p k z k (9.1) 2 k Z ϕ(λ) = ϕ(λ/2)h(e iλ/2 ). (9.2) G(z) G(z) = zh ( z) 8.8 ψ(x) = k Z( 1) k p 1 kϕ(2x k) (8.13 ) Fourier ψ(λ) = ϕ(λ/2)g(e iλ/2 ). (9.3)

248 242 9 Daubechies F[ϕ(x)](λ) = k 3.18 Fourier (1),(6),(7) p k F[ϕ(2(x k/2))](λ) = 1 p k ϕ(λ/2)e ikλ/2 = 2 ϕ(λ/2)h(e iλ/2 ). 8.1 MRA ϕ (8.1) Fourier H(z) 9.2 ϕ ϕ (x k)ϕ(x l)dx = δ kl 2- ϕ(x) = k p kϕ(2x k) H(z) = 1/2 k p kz k H(z) 2 + H( z) 2 = 1, z C, z = 1. (9.4) k 8.1 ϕ(λ + 2πk) 2 = 1 2π, λ R (8.1 ) k Z 9.1 ϕ(λ) = ϕ(λ/2)h(e iλ/2 ) (9.2 ) 1 2π = ϕ(λ + 2πk) 2 k Z = ϕ(λ + (2l)π) 2 + ϕ(λ + (2l + 1)π) 2 l Z l Z = Ä ϕ(λ/2 + (2l)π)H(e i(λ/2+2lπ) ) 2 + ϕ(λ/2 + (2l + 1)π)H(e i(λ/2+(2l+1)π) ) 2 l Z = H(e iλ/2 ) 2 l Z ϕ(λ/2 + 2πl) 2 + H( e iλ/2 ) 2 l Z ϕ((λ/2 + π) + 2πl) /2π 1 = H(e iλ/2 ) 2 + H( e iλ/2 ) 2.

249 9.1 Fourier Haar 2- p 0 = p 1 = 1, p k 0,1 = ψ 9.4 ϕ ϕ (x k)ϕ(x l)dx = δ kl 2- ϕ(x) = k p kϕ(2x k) ψ(x) = k q kϕ(2x k) G(z) = k q kz k ψ (x k)ϕ(x l)dx = 0, kl Z H(z)G (z) + H( z)g ( z) = 0, z = S ï ò H(z) H( z) S = G(z) G( z) SS = ï ò [0, 1) 1 Haar ϕ(x) Fourier ϕ(λ) = 1 1 e ixλ dx = e iλ 1 2π i 2πλ 0 ϕ 2- p 0 = p 1 = 1 H(z) = (1 + z)/2 H(e iλ/2) ϕ(λ/2) = 1 2 (1 + e iλ/2 ) e iλ/2 1 i 2πλ/2 = e iλ 1 i 2πλ = ϕ(λ). ϕ(λ) = H(e iλ/2) ϕ(λ/2) 9.1 H(z) 2 + H( z) 2 = 1 + z z 2 4 z = 1

250 244 9 Daubechies 9.2 Haar ψ(x) = ϕ(2x) ϕ(2x 1) Fourier ψ(λ) = (e iλ/2 1) 2 i. 2πλ G(z) = (1 z)/2 ϕ(λ/2)g(e iλ/2 ) = e iλ 1 i 2πλ 1 e iλ/2 2 = ψ(λ) MRA ϕ 2- (8.10) (9.1) H(z) 2 + H( z) 2 = 1, z = 1 (9.4 ) ϕ(x) = p k ϕ(2x k) (8.10 ) k Z ϕ MRA Daubechies H [8] ϕ ϕ(λ) = H(e iλ/2 ) ϕ(λ/2) = H(e iλ/2 )H(e iλ /2 2 ) ϕ(λ/2 2 ) ( n ) = H(e iλ/2m ) ϕ(λ/2 n ) m=1

251 ( ) ϕ(λ) = H(e iλ/2m ) ϕ(0) m=1 = 1 2π ( m=1 H(e iλ/2m ) ϕ(0) = 1/ 2π ) 9.7 ( ) 9.2 H H(1) = 1 ϕ ) Haar 1 0 x < 1, ϕ 0 (x) = 0 ϕ 1 (x) = k p kϕ 0 (x),... ϕ n ϕ n (x) = k Z p k ϕ n 1 (2x k) (9.5) ϕ(x) = lim n ϕ n (x) 9.8 H(z) = 1 2 (1) H(1) = 1. p k z k k Z (2) H(z) 2 + H( z) 2 = 1, z = 1. (3) H(e iw ) > 0, w π/2. ϕ 0 (x) Haar ϕ n (x) = k p kϕ n 1 (2x k), n 1 {ϕ n (x)} n 1 L 2 ϕ ϕ ϕ = 1/ 2π ϕ (x m)ϕ(x n) dx = δ mn 2- ϕ(x) = k p kϕ(2x k)

252 246 9 Daubechies (9.5) n = 1 Fourier (9.2) ϕ 1 (λ) = ϕ 0 (λ/2)h(e iλ/2 ). (9.2 ) ϕ 0 (0) = 1/ 2π, H(1) = 1 ϕ 1 (0) = 1/ 2π k Z ϕ 1 (λ + 2πk) 2 = k Z H(e iλ/2+iπk) ) 2 ϕ 0 (λ/2 + πk) 2 = H(e iλ/2+i2πl) ) 2 ϕ 0 (λ/2 + 2πl) 2 l Z + H(e iλ/2+iπ(2l+1) ) 2 ϕ 0 (λ/2 + π(2l + 1)) 2 l Z = H(e iλ/2 ) 2 l Z ϕ 0 (λ/2 + 2πl) 2 + H( e iλ/2 ) 2 ϕ 0 (λ/2 + 2πl) 2 l Z = 1 Ä H(e iλ/2 ) 2 + H( e iλ/2 ) 2ä 8.1 (8.1) 2π = 1 2π 8.1 ϕ 1 ϕ n Haar 2- p 0 = p 1 = 1 H(z) = (1 + z)/2 9.8 ϕ 1 ϕ 1 (x) = p 0 ϕ(2x) + p 1 ϕ(2x 1) = ϕ 0 (x) ϕ 0 = ϕ 1 = ϕ 2 =... Haar ϕ H(z) = k p kz k Daubechies[8] 2- p 0 = 1 + 3, p 1 = 3 + 3, p 2 = 3 3, p 3 = ϕ n (x) = k p kϕ n 1 (2x k) ϕ 1, ϕ 2, ϕ 3, ϕ 4, ϕ 5, ϕ 6 9.1

253 (1) (2) (3) ) (5) -1 (6) Daubechies ϕ 2- p 0 = 1 + 3, p 1 = 3 + 3, p 2 = 3 3, p 3 = 1 3 Haar ϕ 0(x)(0) ϕ n(x) = l pkϕn 1 ϕ1, ϕ2, ϕ3, ϕ4, ϕ5. lim n ϕ n(x) ϕ(x)

254 248 9 Daubechies ϕ(x) = l Z p k ϕ(2x l) (8.10 ) ϕ Fourier (9.1) H m 0 (ξ) = H(e iξ ) = 1 p l e ilξ (9.6) 2 l Z m 0 (ξ) 9.2 (9.4) m 0 (ξ) 2 + m 0 (ξ + π) 2 = 1 (9.7) (9.2) ϕ(ξ) = m 0 (ξ/2) ϕ(ξ/2). (9.2 ) ϕ ψ(x) = l Z( 1) l p 1 lϕ(2x l) (8.13 ) Fourier (9.3) ψ(ξ) = ϕ(ξ/2) Ä e iξ/2 H ( e iξ/2 ) ä = e iξ/2 m 0(ξ/2 + π) ϕ(ξ/2). (9.8) 9.11 ( ) ϕ(x) {p k } MRA ψ(x) (8.13) N N (1) ψ k x k ψ(x) dx = 0 0 k N 1. (9.9)

255 (2) m 0 π k m (k) 0 (π) = 0 0 k N 1. (9.10) (3) m 0 (ξ) 2π L m 0 (ξ) = Ç å 1 + e iξ N L(ξ) (9.11) 2 (4) p l ( 1) l l k = 0 (9.12) l Z (1) (2) (9.8) ϕ(0) = 1/ 2π ψ(0) = m 0(π) ϕ(0) m 0 (π) = 0 Fourier (3.11) Å d dξ ψ(0) i = m ) (π) 2 ϕ(0) 1 ã d 2 dξ ϕ(0) + 1 d 2 dξ m 0(π) ϕ(0) = 0 d dξ m 0(π) = 0 0 k N 1 ψ (k) (0) = 0 m (k) 0 (π) = 0 (2) (3) {p k } Z- H(z) = k p k z k m 0 (ξ) = H(e iξ ) H(z) z 1 J z J H(z)

256 250 9 Daubechies H(z) H( 1) = H(e iπ ) = m 0 (π) (2) H (k) ( 1) = 0 0 k N 1 Taylor H(z) = N 1 k=0 H (k) ( 1) (z 1 + 1) k + 1 k! N! (z 1 + 1)R N (z) = 1 N! (z 1 + 1)R N (z) m 0 (ξ) = He iξ = L(ξ) = 2N N! R N(e iξ ) Ç å 1 + e iξ N L(ξ) 2 R N (z) z 1 L(ξ) 2π (3) (3) z 1 a(z) H(z) = (z 1 + 1) N a(z) H(z) z = 1 N z 0 H(z) r z 0 = e iξ0 ξ 0 m 0 (ξ) r z = 1 H(z) N π m 0 (ξ) N (2) (2) (4) m 0 (ξ) {p k } m (k) 0 (ξ) = 1 2 p l ( il) k e ilξ. l m (k) ( i)k 0 (π) = 2 l p l l k (e iπ ) l. = ( i)k 2 p l l k ( 1) l l

257 9.4 Daubechies (a) (b) Daubechies N = p 0 = 1 + 3, p 1 = 3 + 3, p 2 = 3 3, p 3 = 1 3 : (a) ϕ(x) 9.1, (b) ψ(x) = k ( 1)k p 1 kϕ(2x k). 9.4 Daubechies Daubechies Haar ϕ ψ [8][9]

258 252 9 Daubechies 9.11(3) M 0 (ξ) = m 0 (ξ) 2 = 1 + e ixi 2 2N L(ξ) 2 = cos 2N (ξ/2)l(ξ) (9.13) L(ξ) = L(ξ) 2 L(ξ) L(ξ) = l 0 + (l n e iξ ) + l n e iξ ) = l 0 + 2l n cos ξ n=1 cos ξ = 1 2 sin 2 ξ/2 L(ξ) sin 2 /xi/2 P (sin 2 ξ/2) M 0 (ξ) = Å cos 2ã ξ 2N Å P sin 2 ξ ã 2 (9.7) 1 = M 0 (ξ) + M 0 (ξ + π) Å = cos 2ã ξ 2N Å P sin 2 ξ 2 = ã Å + Å cos 2ã ξ 2N Å P sin 2 ξ ã Å + sin ξ 2N P 2 2ã n=1 cos ξ + π ã 2N Å P sin 2 ξ + π ã 2 2 Å cos 2 ξ ã. 2 y = sin 2 ξ/2 Daubechies (1 y) N P (y) + y N P (1 y) = 1. (9.14) m 0 (ξ) 2 + m 0 (ξ + π) 2 = 1 (9.7) ξ 0 sin 2 ξ/2 1 P P (y) 0, 0 y 1. (9.15) ϕ ψ (9.14) (9.15) P (y) *1 Daubechies [8] *1 P (y) m 0 (ξ) 2 m 0 [9, 6.3.6]

259 9.5 Daubechies Daubechies Daubechies 9.12 N 1 (N 1) Daubechies P N 1 (y) = N 1 k=0 Ç å 2N 1 y k (1 y) N 1 k (9.16) k Daubechies (9.14) Daubechies 1 = y N P N 1 (1 y) + (1 y) N P N 1 (y) N N ( ) 2N 1 2N 1 Ç å 2N 1 1 = ( y) + y = (1 y) k y 2N 1 k k k=0 N 1 Ç å 2N 1 Ç å 2N 1 2N 1 = (1 y) k y 2N 1 k + (1 y) k y 2N 1 k k k k=0 k=n N 1 Ç å N 1 Ç å 2N 1 2N 1 = (1 y) k y 2N 1 k + (1 y) 2N 1 m y m k m k=0 m=0 N 1 Ç N 1 Ç å 2N 1 2N 1 = y å(1 N y) k y 2N 1 k + (1 y) N (1 y) N 1 m y m. k m k=0 m= N = 1 P 0 (y) = 1 N = 2 P 1 (y) = 1 + 2y N = 3 P 2 (y) = 1 + 3y + 6y 2 N = 4 P 3 (y) = 1 + 4y + 10y y 3.

260 254 9 Daubechies N Daubechies P N 1 (y) 9.11 (3) (9.11) m 0 (ξ) = 1 p k e ikξ 2 k Z Ç å 1 + e iξ N = L(ξ) (9.11 ) 2 {p k } (9.13) L(ξ) 2 = L(ξ) Å = P N 1 sin 2 ξ ã 2 m 0 (ξ) {p k } ϕ ψ ψ(x) = k Z( 1) k p 1 kϕ(2x k) Daubechies P N 1 (y) N 2N 1 P 2N 1 (y)(1 y) N P N 1 (y) (9.17) N = 1 P 0 (y) = 1 Å L(ξ) 2 = P 0 sin 2 ξ ã = 1 2 L(ξ) = 1

261 9.5 Daubechies 255 Ç å 1 + e iξ m 0 (ξ) = = 1 p k e ikξ 2 2 k Z {p k } 1 k = 0, 1 p k = Haar N = 2 P 1 (y) = 1 + 2y Å L(ξ) 2 = P 1 sin 2 ξ ã = sin 2 ξ = 2 cos ξ. 2 2 L(ξ) L(ξ) = a + be iξ L(ξ) 2 = (a 2 + b 2 ) + 2ab cos ξ a 2 + b 2 = 2, 2ab = 1 a, b m 0 (0) = L(0) = 1 a + b = 1 a + b = 1, a b = 3 a = , b =

262 256 9 Daubechies m 0 (ξ) = Ç å 1 + e iξ 2 Ç = p 0 = = 1 2 pk e ikξ,, p 1 = å 3 e iξ 2 e iξ e 2iξ e 3iξ 8, p 2 = 3 3, p 3 = N = 3 P 2 (y) = 1 + 3y + 6y 2 Å L(ξ) 2 = P 2 sin 2 ξ ã = sin 2 ξ ξ sin4 2 = cos ξ + 3 cos 2ξ. 4 L(ξ) L(ξ) = a + be iξ + ce 2iξ L(ξ) 2 = (a 2 + b 2 + c 2 ) + (2ab + 2bc) cos ξ + 2ac cos 2ξ a 2 + b 2 + c 2 = 19 4, 2ab + 2bc = 9 2, 2ac = 3 4. a, b.c (a b + c) 2 = (a 2 + b 2 + c 2 ) (2ab + 2bc) = 2ac = 10 a + b + c = 1 (a b + c) 2 = (a b + c 2b) 2 = (1 2b) 2

263 9.5 Daubechies 257 b = 1 2 (1 ± 10) b = (1 10)/2 a + c = 1 b = 1 2 (1 + 10), ac = 3 8. a = 1 Å 1 +» 10 ± ã 10 4 b = 1 2 (1 10) c = 1 4 Å » ã 10. m 0 (ξ) = Ç å 1 + e iξ 3 (a + be iξ + ce i3ξ) 2 p 0 = 1 Å 1 +» 10 ± ã p 1 = 1 Å 5 +» 10 ± ã p 2 = 1 Å 5» 10 ± ã 10 8 p 3 = 1 Å 5» ã 10 8 p 4 = 1 Å 5 +» ã p 5 = 1 Å 1 +» ã

264

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