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2 F ( ) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 24 ( ) A α a alpha ǽlf@ B β b beta bí:t@, béit@ Γ γ g gamma gǽm@ δ d delta délt@ E ϵ, ε e epsilon épsil@n/-lan, epsáil@n Z ζ z zeta zí:t@ H η e eta í:t@, éit@ Θ θ, ϑ t theta Tí:t@, Téit@ I ι i iota íout@, aióut@ K κ k kappa kǽp@ Λ λ l lambda lǽmd@ M µ m mu mju:, mu: N ν n nu nju:, nu: Ξ ξ x xi gzai, ksi:/-sai O o o omicron óumikr@n, oumái- Π π, ϖ p pi pai P ρ, ϱ r rho rou Σ σ, ς s sigma sígm@ T τ t tau tau, to: Υ υ u upsilon jú:psil@n, ju:psáil@n Φ ϕ, φ p phi fi:, fai X χ c chi kai Ψ ψ p psi psai, psi:/-sai Ω ω o omega óumig@, oumég@/-mí:- ξ ρ ( ζ ) 1
3 N = {1, 2, } N 0 = {0, 1, 2, } ( Z 0 ) Z = {0, ±1, ±2, } Q = R = C = K R, C X = {f f : R C } (, p. 19) δ mn (m = n 1, 0) z z ( 1 + 2i = 1 2i) Re z, Im z z ( Re(1 + 2i) = 1, Im(1 + 2i) = 2) Span φ 1,..., φ N = {c 1 φ c n φ n c 1,, c n K} (p. 130) f Fourier F f, Ff, f, F[f(x)](ξ) F, F F ( F ) sinc x := sin x x ( Wikipedia [sínk]) a a e ibx dx = 2a sinc(ab). f def. ( x) f( x) = f(x). f def. ( x) f( x) = f(x). f C 1 f {a n } n N a: N C, a(n) = a n C N ( X Y Y X ) {a n } n Z a: Z C C Z f : [a, b] C {x j } N j=0 a = x 0 < x 1 < < x N = b, j {1, 2,, N} f (x j 1, x j ) C 1 lim f(x), x x j 1 +0 lim f(x), x x j 0 lim f (x), x x j 1 +0 lim f (x) x x j 0 f : [a, b] C {x j } N j=0 a = x 0 < x 1 < < x N = b, j {1, 2,, N} f [x j 1, x j ] C 1 x j f (x j 0), f (x j + 0) 2
4 Fourier ( + ) ( ) ( ) ( = ) Fourier ( ) Bessel, Parseval, : Fourier Fourier Fourier, Fourier Fourier Fourier e ax2 Fourier : Fourier x 2 + a Mathematica Fourier Fourier Fourier Fourier Fourier ( ) : ( ) Fourier Fourier Fourier
5 3.3 : FFT Fourier Mathematica Fourier : : Fourier C n, A n, B n Fourier guitar-5-3.wav Fourier PCM C n (1 n N 1) n n /T Mathematica Fourier Fourier Fourier Fourier Fourier Dirichlet f g = g f (f g) h = f (g h) (the Titchmarsh convolution theorem) Fourier Fourier Fourier Fourier Fourier Fourier Fourier
6 7.5.4 Fourier Fourier ( ) Fourier (LTI ) FIR Fourier : : Fourier : CT A 124 B & 125 C Fourier ( 1 ) 127 C C.2 Fourier, Fourier C.3 : (Bessel, Schwarz, ) C.4 Fourier C D Hilbert 137 D D.2 Riesz D E Fourier 140 E.1 Lebesgue E E E
7 E.1.4 Lebesgue, Lebesgue E E.1.6 Lebesgue E.2 Lebesgue E E E.3 Lebesgue Fourier E.4 Lebesgue Fourier F ( ) 146 F F.2 R Fourier F.2.1 L 1 (R) F.2.2 L 2 (R) F.2.3 Fourier F F.2.5 Fourier F.2.6 Fourier F F.3 R Fourier F.4 Z Fourier G 152 G.1 1 R G.1.1 d Alembert G.1.2, d Alembert G H [0, ) Laplace 156 I 157 I I.2 sinc I.3 : J Laplace z 160 J.0.1 Laplace J.1 z
8 Fourier ( 1 ) 2 Fourier Lebesgue 3 4 Fourier ( ) 2 ( ) 3 2 ( ) ( Fourier ) ( ) Mathematica ( ) 7
9 Fourier Fourier (Jean Baptiste Joseph Fourier, 1768 Auxerre 1830 Paris ) Fourier [1] (1809, 1812, 1822) ( ) c u t = k u ( [2] 2 1, 2) u = u(x, t) x, t c ( ) k ( ) : n 2 u u =. x 2 j j=1 Fourier (Fourier, Fourier, Fourier ) Fourier 1 2 u c 2 t 2 = u ( ) 18 Fourier ( [3], 91) Fourier ( ) Shannon 20 Fourier Shannon Claude Elwood Shannon (1916 Gaylord 2001 Medford ) [4] (1948 ) (1949 ) FFT FFT ( Fourier, fast Fourier transform) Fourier ( ) 1965 Cooley-Tukey [5] 8
10 ( ) FFT Gauss (Johann Carl Friedrich Gauss, ) ( ) Cooley- Tukey ( ) Cooley-Tukey [5] : ( ) ( ) ( ) (,,,, ) ( ) ( ) ( ) ( L 2 ) ( ) ( ) Lebesgue n Z sin nπ = 0, cos nπ = ( 1) n, sin (n + 1/2) π = ( 1) n. b a b f (x)g(x) dx = [f(x)g(x)] b a f(x)g (x) dx. : f : R C f(x) dx = R2 lim f(x) dx. R 1,R 2 + R 1 a 9
11 ( ):, a, b, α, β R, a < b, α < β, F : [a, b] [α, β] C C 1 d dξ b a F (x, ξ) dx = b a F (x, ξ) dx ξ (ξ [α, β]). F (x, ξ) ξ φ(x), φ(x) dξ < + φ d dξ F (x, ξ) dx = F (x, ξ) dx ξ (ξ [α, β]). (Fourier F (x, ξ) = f(x)e ixξ φ(x) := xf(x) + ) (Gauss ) e x2 dx = π. φ(x) dx < 10
12 1 Fourier ( + ) Fourier 1.1 ( ) 2 (?) ( ) f : R C (1.1) (1.2) a n := 1 π b n := 1 π f(x) cos nx dx (n = 0, 1, 2, ), f(x) sin nx dx (n = 1, 2, 3, ) {a n } n 0, {b n } n 1 ( ) a 0 (1.3) 2 + a 0 n (a n cos nx + b n sin nx) := lim n 2 + (a k cos kx + b k sin kx) n=1 k=1 (x R) f(x) (1.4) f(x) = a (a n cos nx + b n sin nx) (x R). n=1 {a n }, {b n } f Fourier (1.3) f Fourier (1.4) f Fourier Euler e iθ = cos θ + i sin θ cos θ = eiθ + e iθ 2, sin θ = eiθ e iθ, 2i cos( θ) = cos θ, sin( θ) = sin θ 11
13 1.1.2 ( ) f : R C (1.5) c n := 1 f(x)e inx dx {c n } n Z (1.6) n= c n e inx := lim n n k= n c k e ikx (x R) f(x) (1.7) f(x) = c n e inx (x R). n= {c n } f ( ) Fourier (1.6) f ( ) Fourier (1.7) f ( ) Fourier f Fourier Fourier ( Fourier 5, 6 ) 1. a n, b n, c n (1.1), (1.2), (1.5) (1) n N c n = 1 2 (a n ib n ), c n = 1 2 (a n + ib n ). c 0 = a 0 2. (2) n N a n = c n + c n, b n = i (c n c n ). a 0 = 2c 0. (3) n N a 0 n 2 + (a k cos kx + b k sin kx) = k=1 n k= n c k e ikx. (4) f a n b n c n = c n ( c 0 ). a n = 2 Re c n, b n = 2 Im c n. T (cos ( 2nπ x), sin ( 2nπ x) e i 2nπ T x ) T T ( ) : cos nx α+ cos nx ( α R) f(x) sin nx e dx = f(x) sin nx inx α e dx. inx [, π] [0, ] ( ) [, π] 12
14 (, π] f f(x) := f(y) (x R y x y (mod ) y (, π]) f ( f f ) f(x) = a (a n cos nx + b n sin nx) (x R), a n = 1 π b n = 1 π n=1 f(x) cos nx dx (n = 0, 1, 2, ), f(x) sin nx dx (n = 1, 2, 3, ) (, π] f f a n, b n f f f(x) = a (a n cos nx + b n sin nx) (x (, π]) n=1 f Fourier cos, sin : f Fourier = a a n cos nx, f Fourier = n=1 b n sin nx, n=1 b n = 2 π a n = 2 π 0 0 f(x) cos nx dx, f(x) sin nx dx. ( ) a a ( )dx = 0, a a ( )dx = 2 a 0 ( )dx ( ) Fourier f ( Lebesgue ) (1.8) lim n a n = lim n b n = 0 (Riemann-Lebesgue, ) Fourier 0 (1.8) (1.8) ( (1.24)) f C k lim n n k a n = lim n n k b n = 0 ( ) Fourier n k ( a n + b n ) < f C k n=1 13
15 1.1.3 ( Fourier ) f : R C, g : R C f(x) = x 2, g(x) = 2x ( x < π) f g Fourier ( ) f(x) = π2 3 4 ( ) n 1 cos nx ( 1) = π2 cos x n cos 2x cos 3x n=1 ( ) n 1 sin nx sin x sin 2x sin 3x g(x) = 4 ( 1) = 4 + (x R). n n=1 f f Fourier s n f0[x_]:=x^2 f[x_]:=f0[mod[x,2pi,-pi]] Plot[f[x],{x,-3Pi,3Pi}] s[n_,x_]:=pi^2/3-4sum[(-1)^(k-1)cos[k x]/k^2,{k,1,n}] Plot[s[10,x],{x,-3Pi,3Pi}] Manipulate[Plot[{f[x],s[n,x]},{x,-3Pi,3Pi}],{n,1,20}] (x R), (Mod[a,b,c] a b r ( c r < c + b ) ) f f n : f 1.2: s 10 s n f g g Fourier s n g0[x_]:=2x g[x_]:=g0[mod[x,2pi,-pi]] Plot[g[x],{x,-3Pi,3Pi}] sg[n_,x_]:=4sum[(-1)^(k-1)sin[k x]/k,{k,1,n}] Plot[sg[10,x],{x,-3Pi,3Pi}] Manipulate[ Plot[{g[x],sg[n,x]},{x,-3Pi,3Pi},PlotPoints->200,PlotRange->{-8,8}], {n,1,50,1}] 14
16 1.3: n f s n : g 1.5: s 10 15
17 1.6: n g s n ( g Fourier, PlotPoints->200 ) g C 1 D := {(2k 1)π k Z} g g R \ D C 1 x R \ D g(x) g(x + 0) + g(x 0) + ( ) x D = = n g f (x = (2k 1)π, k Z) g(x + 0), g(x 0) ( ) n n 0 Gibbs 1.2 (2017 ) Fourier s n (x) := a n (a k cos kx + b k sin kx) = k=1 n k= n c k e ikx {s n } f 2 16
18 Fourier x 1 f ( ) ( 1.5) 3 (1) ( ) ( ) ( x R) lim s n (x) = f(x) n {s n } f ( x {s n (x)} f(x) ) (2) (L p ) (p ). p 1 p < lim n s n (x) f(x) p dx = 0 {s n } f L p s n f ( ) 0 L p p p = 1 s n (x) f(x) p dx = s n f p p lim s n f n p = 0 ( x π) s n (x) f(x) p dx y = f(x) y = s n (x) p = 2 ( ) (3) ( ) lim sup s n (x) f(x) = 0 n x R {s n } f sup ( ) 1 1 R N ( ) 17
19 ( p [1, ) ) L p ( ) p ( p s n (x) f(x) p dx sup s n (x) f(x) dx = π sup s n (x) f(x) ) 0 (n ), x R x R ( x 0 R) s n (x 0 ) f(x 0 ) sup s n (x) f(x) 0 (n ). x R f : R C C 1 f Fourier {s n } f : lim sup f(x) s n (x) = 0. n x R ( ) f : R C C 1 f Fourier {s n } f L 2 : lim n f(x) s n (x) 2 dx = 0. C 1 Lebesgue 2 L 2 (?) Lebesgue ( 2, ) ( ) 1.3, 1.4 ( ) L f : R C C 1 x R lim s n(x) = n f(x) f(x + 0) = f(x + 0) + f(x 0) 2 3 (x f ) (x f ). lim f(y), f(x 0) = lim f(y). y x+0 y x 0 Gibbs (Gibbs [6], [7]) f Fourier ( ) a n (z c) n ( ) (1.3) 2 f Lebesgue n=0 f(x) 2 dx <
20 (Taylor ) Fourier ( ) 1.3 ( ) Fourier ( ) (1.9) m, n Z 0, m n m, n N, m n m Z 0, n N cos mx cos nx dx = 0, sin mx sin nx dx = 0, cos mx sin nx dx = 0. (1.10) m, n Z, m n e imx e inx dx = 0. ( 1 + 2i = 1 2i. e inx = cos(nx) + i sin(nx) = cos(nx) i sin(nx) = e inx ) m, n Z, m n cos mx cos nx dx = 0 (1.11) X := {f f : R C } X 5 X C ( ) f, g X (1.12) (f, g) = (f, g) L 2 := f(x)g(x) dx 5 f, g X (f + g)(x) := f(x) + g(x) (x R) f + g : R C f + g X. f X, λ C (λf)(x) := λ f(x) λf : R C λf X. 19
21 f g L 2 f X (1.13) f = f L 2 := (f, f) = f(x) 2 dx f L m, n Z 0, m n (cos mx, cos nx) = 0. m N π cos mx = cos 2 mx dx = π sin mx = sin 2 mx dx = 1 + cos 2mx dx = π, 2 1 cos 2mx dx = π, 2 π cos(0x) = cos 2 (0x) dx = dx = ( ) X = X (1.12) (, ) = (, ) L 2 (i), (ii), (iii) (i) f X (f, f) 0. f = 0 (ii) f, g X (g, f) = (f, g). (iii) f 1, f 2, g X, c 1, c 2 C (c 1 f 1 + c 2 f 2, g) = c 1 (f 1, g) + c 2 (f 2, g). (i) z z z = z 2 (ii) (f, f) = f(x)f(x) dx = f(x) 2 dx 0. (f, f) = 0 f(x) 2 = 0 f(x) = 0. ( x x f(x) = 0 x 0 0 Lebesgue ) (f, g) = f(x)g(x) dx = f(x)g(x) dx = f(x) g(x) dx = g(x)f(x) dx = (g, f). 20
22 (iii) (c 1 f 1 + c 2 f 2, g) = (c 1 f 1 (x) + c 2 f 2 (x)) g(x) dx = c 1 f 1 (x)g(x) dx + c 2 f 2 (x)g(x) dx = c 1 (f 1, g) + c 2 (f 2, g) (, ) C X X X (, ) (i), (ii), (iii) X (, ) X (i) f X (f, f) 0. f = 0 (ii) f, g X (g, f) = (f, g). (iii) f 1, f 2, g X, c 1, c 2 C (c 1 f 1 + c 2 f 2, g) = c 1 (f 1, g) + c 2 (f 2, g). (i), (ii), (iii) ( ) (f, g) f g f, g (f g) (, ) f := (f, f) 3 (i) f X f 0. f = 0 (ii) f X, λ C λf = λ f. (iii) f, g X f + g f + g. 4. (ii), (iii) (a) f, g 1, g 2 X c 1, c 2 C (f, c 1 g 1 + c 2 g 2 ) = c 1 (f, g 1 ) + c 2 (f, g 2 ). (b) f, g X f + g 2 = f Re (f, g) + g 2. (Re Re(1 + 2i) = 1.) X C N C N C C N ( ) (x, y) = N x j y j j=1 21
23 X (e inx cos nx, sin nx ) (, ) (f, g) = f(x)g(x) dx (i), (ii), (iii) ((ii), (iii) ) R (i) f X (f, f) 0. f = 0 (ii) f, g X (g, f) = (f, g). (iii) f 1, f 2, g X, λ 1, λ 2 R (λ 1 f 1 + λ 2 f 2, g) = λ 1 (f 1, g) + λ 2 (f 2, g) (R, ) R X (i), (ii), (iii) (, ) X R (, ) X (i), (ii), (iii) R N R ( ) C N ( ) ( ) (x, y) = N x j y j j=1 (i) (x, x) 0 ( ) ( ) ( ) ( ) X X C R ( X = C n X = R n ) ( ) X a, b X a b ( (a, b) = 0) a + b 2 = a 2 + b 2. ( 2 = = ) 22
24 a + b 2 = (a + b, a + b) = (a, a) + (a, b) + (b, a) + (b, b) = (a, a) + (a, b) + (a, b) + (b, b) = a b 2 = a 2 + b X φ n (n = 1,, N) 2 (n m (φ n, φ m ) = 0) N 2 φ n = n=1 ( ) N φ n 2 n=1 R n C n (x, y) x y (Schwarz ) (Schwarz ) X C R f, g X (f, g) f g. ( f g 1 ) ( ) f g 1 f g 1 λ C λf + g 0 0 < λf + g 2 = λ 2 f Re λ(f, g) + g 2. (f, g) = (f, g) e iθ θ R t λ = te θ λ(f, g) = t (f, g), Re λ(f, g) = t (f, g). 0 < t 2 f 2 + 2t (f, g) + g 2 (t R). t 2 (f, g) 2 f 2 g 2 < 0. (f, g) < f g (R ( ) C λ = te iθ ) 23
25 6. ( ) (i) (i ) f X (f, f) 0. ( (f, f) = 0 f = 0 ) Schwarz (, ) X C R {φ n } X (1) {φ n } X (i), (ii) (i) ( n, m) n m (φ n, φ m ) = 0. (ii) ( n) (φ n, φ n ) 0. (2) {φ n } X (φ m, φ n ) = δ mn ({φ n } (i) (ii) (ii) ) δ mn (1.14) δ mn = { 1 (m = n) 0 (m n). Kronecker X = C N, e n = n 1 0 N, {e n } N n=1 X Gram-Schmidt ( ) {φ n } n X ψ n := 1 φ n φ n {ψ n } n X ( ( n φ n 0 )) ( {ψ n } {φ n } ) 24
26 ( 1 (ψ m, ψ n ) = φ m φ m, 1 = = δ mn. ) 1 φ n φ n = 1 1 φ m φ n (φ m, φ n ) 0 = 0 φ m φ n (m n) 1 φ m φ m φ m 2 = 1 (m = n) ( Fourier ) {1, cos x, sin x, cos 2x, sin 2x,, cos kx, sin kx, } X = X ( k N) cos kx = π, sin kx = π. cos(0x) = 1 =. { 1, cos x, sin x } cos 2x sin 2x cos kx sin kx,,,,, π π π π π π X { 1, e ix, e ix, e 2ix, e 2ix,, e ikx, e ikx, } X { 1 1, 1 e ix, 1 e ix, 1 e 2ix, 1 e 2ix,, 1 e ikx, } 1 e ikx, X ( ) X f = n c n φ n (1) {φ n } ( n) c n = (f, φ n) (φ n, φ n ). (2) {φ n } ( n) c n = (f, φ n ). f = C.1 C.1.2 N c n φ n n=1 25
27 (1) n ((φ m, φ n ) m = n 0 ) ( N ) N (f, φ n ) = c m φ m, φ n = c m (φ m, φ n ) = c n (φ n, φ n ). m=1 m=1 (φ n, φ n ) ( 0) c n = (f, φ n) (φ n, φ n ). (2) (f, φ n ) = c n (φ n, φ n ) (1) (φ n, φ n ) = 1 c n = (f, φ n ) (Fourier ) n N ( ) a n = (cos nx, cos nx) = f(x) = a (a n cos nx + b n sin nx) n=1 cos 2 nx dx = 1 2 (1 + cos 2nx)dx = 1 2 = π (f, cos nx) (cos nx, cos nx) = 1 f(x)cos nx dx = 1 f(x) cos nx dx. π π b n a 0 /2 cos(0x) = 1 a 0 2 = (f, cos(0x)) (cos(0x), cos(0x)) = 1 f(x) cos(0x)dx. a 0 = 1 π f(x) cos(0x) dx ( Fourier ) X = X, φ n (x) = e inx (φ n, φ n ) = f = c n φ n n= (n Z) c n = (f, φ n) (φ n, φ n ) = 1 f(x)e inx dx = 1 f(x)e inx dx ( T Fourier ) T f Fourier f(x) = a ( a n cos 2nπx + b n sin 2nπx ) T T n=1 a n, b n n N ( cos 2nπx T, cos 2nπx ) T/2 = cos 2 2nπx T/2 T T/2 T dx = 1 + cos 4nπx T dx = T T/2 2 2, ( sin 2nπx T, sin 2nπx ) = T T/2 T/2 sin 2 2nπx T/2 T dx = T/ cos 4nπx T dx = T 2 2.
28 n = 0 a n = b n = ( (f, cos 2nπx ( ) T cos 2nπx, cos ) = 2 2nπx T T T (f, sin 2nπx ( ) T sin 2nπx, sin ) = 2 2nπx T T T cos 2nπx T T/2 T/2 T/2 T/2, cos 2nπx ) = (1, 1) = T f(x) cos 2nπx T dx, f(x) sin 2nπx T dx, T/2 T/2 dx = T a 0 2 = (f, 1) (1, 1) = 1 T a 0 = 2 T T/2 T/2 T/2 T/2 f(x)dx. f(x)dx. 1.4 ( = ) ( ) Fourier ( ) Fourier ( ) ( ) (, ) ( ) f f = (f, f) Span φ 1,, φ N φ 1,..., φ N : { N } (1.15) Span φ 1,, φ N = φ 1,..., φ N = c n φ n c 1,..., c N K. K = R K = C. n= ( ) V f, V g, f V V h. V f, V g, f V V h. g f h 27
29 ( h) φ 1,..., φ N V h = N n=1 (f, φ n ) (φ n, φ n ) φ n ( h = N (f, φ n )φ n ). h f V ( ) ( f V ) n=1 (1) (f h) V h V 6 f h = inf f g ( ). g V (2) f h = inf f g ( ) h V (f h) V. g V (1) (1) g V f g 2 = f h 2 + g h 2 7 f g f h. f h (2) ( ) I[g] := f g 2 (g V ) I g = h v V t K h + tv V I[h + tv] I[h] F (t) := I[h + tv] (t K) F t = 0 F (t) = I[h + tv] = f (h + tv) 2 = (f h) tv 2 = f h 2 2 Re [t (f h, v)] + t 2 v 2 6 ( ) ( ) D.1 7 g, h V g h V (f h) (g h) 28
30 F t = 0 (f h, v) = 0 K = R K = C ( K = R ) (i) K = R ( Re t 2 = t 2 ) 2 F (t) = f h 2 2t (f h, v) + t 2 v 2 t = (f h, v) = 0. (ii) K = C (f h, v) = (f h, v) e iθ (θ R) t = se iθ (s R) F (t) = f h 2 2s (f h, v) + s 2 v 2. s s = 0 (f h, v) = 0. (f h, v) = 0. (f h) V φ 1,..., φ N X V = Span φ 1,, φ N h (f V f V ) h V N h = c n φ n c 1,..., c N (f h) V n (1.16) h = j=1 (f h, φ n ) = 0 ( N ) (f, φ n ) = (h, φ n ) = c m φ m, φ n = c n (φ n, φ n ) N n=1 φ 1,, φ N N (1.17) h = (f, φ n ) φ n n=1 m=1 c n = (f, φ n) (φ n, φ n ). (f, φ n ) (φ n, φ n ) φ n ( ). ( ) ( ) V V (Hilbert ) (, D.1 ) 29
31 1.4.2 Fourier ( ) Fourier ( ) {φ n } f n=1 (f, φ n ) (φ n, φ n ) φ n ( f Fourier ) N (f, φ n ) (φ n, φ n ) φ n h f Fourier f n=1 V = span φ 1, φ 2,, φ N φ := (φ, φ) f s N = inf g V N f g ( ) Fourier ( ) s N := N c n φ n n=1 N f f s N f Fourier s N (x) = a N (a n cos nx + b n sin nx) n=1 f V = Span cos 0x, cos x, sin x, cos 2x, sin 2x,, cos Nx, sin Nx f Fourier f s N (x) = N n= N c n e inx V = Span e i0x, e ix, e ix, e 2ix, e 2ix,, e inx, e inx 30
32 1.4.3 Bessel, Parseval, Fourier Bessel Fourier ( ) Bessel {φ n } ψ n := 1 φ n φ n {ψ n } ( Bessel ) X {ψ n } N ( f X) (f, ψ n ) 2 f 2 n=1 N s N := ( f X) (f, ψ n ) 2 f 2. n=1 N (f, ψ n )ψ n 0, s N, f 3 n=1 s N 2 + f s N 2 = f 2 s N 2 f 2. ( ) {ψ n } ( ) 8 N (f, ψ n ) 2 ψ n 2 f 2. n=1 ( ψ n = 1) N (f, ψ n ) 2 f 2. n=1 N N (f, ψ n ) 2 f 2. n=1 8 N 2 N N j k (ψ j, ψ k ) = 0 c k ψ k = c k ψ k, c j ψ j = N N c k c k (ψ k, ψ k ) = c k 2 ψ k 2. k=1 k=1 k=1 31 k=1 j=1 N k=1 j=1 N c k c j (ψ k, ψ j ) =
33 1.4.4 ( Bessel ) X {φ n } ( f X) N (f, φ n ) 2 φ n 2 f 2 n=1 N (f, φ n ) 2 ( f X) φ n=1 n 2 f 2. ψ n = 1 φ φ n n {cos mx} m Z 0 {sin nx} n N {e inx } n Z f Fourier f (complete) ( ) X {φ n } n N X {φ n } (complete) ( f X) lim f s N(f) = 0 N s N (f) := N n=1 (f, φ n ) (φ n, φ n ) φ n (N N). s N (f) 2 + f s N (f) 2 = f 2 {φ n } lim s N(f) 2 = f 2 N (1.18) (1.19) ( f X) ( f X) (f, φ n ) 2 φ n 2 = f 2 ( ), (f, ψ n ) 2 ( ) n=1 n=1 ( Bessel ) (1.18), (1.19) Parseval Fourier L 2 Lebesgue L 2 ( ) Bessel Fourier c n = (f, φ n) n 0 (φ n, φ n ) Fourier 32
34 1.4.6 (Fourier,, Parseval ) f : R R f (, π) a n = 1 π b n = 1 π c n = 1 (1), (2), (3) f(x) cos nx dx (n = 0, 1, ), f(x) sin nx dx (n = 1, 2, ), f(x)e inx dx (n Z) (1) f(x) M a n, b n 1 π c n 1 f(x) dx, f(x) dx. (2) (Riemann-Lebesgue ) a n 2M (n = 0, 1, ), b n 2M (n = 1, 2, ), c n M (n N). lim a n = lim b n = 0, n n lim c n = 0. n ± (3) (Parseval ) f (, π) ( f 2 ) ( a 0 2 ( π + an 2 + b n 2)) = f 2, 2 n=1 c n 2 = f 2. n= (1) a n = 1 π f(x) cos nx dx 1 π b n c n = 1 π f(x)e inx dx 1 f(x) cos nx dx 1 π f(x) dx. f(x)e inx dx 1 f(x) dx. f(x) M f(x) dx M dx = M. (2) ( ) f 2 Bessel ( (3)) 0 33
35 (3)?? cos nx 2 dx = { π (n N) (n = 0), sin nx 2 dx = π 1.5 f : R C Fourier (1.20) f(x) = a (a n cos nx + b n sin nx) = n=1 c n e inx n= (x R). a n, b n f Fourier, c n f Fourier f a n (f), b n (f), c n (f) a n = a n (f) := 1 f(x) cos nx dx (n Z 0 ), b n = b n (f) := 1 π π c n = c n (f) := 1 f(x)e inx dx (n Z). f(x) sin nx dx (n N), (1.20) ( ) f (x) (1.21) f (x) =?? ( na n sin nx + nb n cos nx) = inc n e inx n=1 n= (x R). (1.20) (1.21) f Fourier f Fourier (1.22) f (x) ( na n sin nx + nb n cos nx) = n=1 n= inc n e inx ( Fourier ) f : R C [, π] C 1 { a n (f nb n (f) (n N) ) =, b n (f ) = na n (f) (n N), 0 (n = 0) c n (f ) = inc n (f) (n Z). f Fourier ( ) ( na n sin nx + nb n cos nx) = inc n e inx. n=1 n= 34
36 f C 1 a n (f ) = 1 f (x) cos nx dx = 1 ([f(x) cos nx] π π π = n 1 { π nb n (f) (n N) f(x) sin nx dx =, π 0 (n = 0) b n (f ) = 1 f (x) sin nx dx = 1 ([f(x) sin nx] π π π = n 1 π c n (f ) = 1 = in 1 f(x) cos nx dx = na n (f), f (x)e inx dx = 1 f(x)e inx dx = inc n (f) ( [f(x)e inx ] π (n Z). ) f(x)( n sin nx)dx ) f(x)(n cos nx)dx f(x) ( ine inx) ) dx f [, π] C 1 {x k } N k=0 f [xk 1,x k ] C 1 c n (f ) = 1 = 1 = 1 = in N = x 0 < x 1 < < x N = π, f (x)e inx dx = ( [f(x)e inx ] x k x k 1 N 1 k=1 xk xk x k 1 f (x)e inx dx f(x) ( ine inx) dx k=1 x k 1 ( ) f(x N )e inx N f(x 0 )e inx 0 + in f(x)e inx dx 1 π f(x)e inx dx = inc n (f). ) f(x) = x 2 ( x < π) Fourier ( ) f(x) = π2 cos 1x 3 4 cos 2x cos 3x g(x) = 2x ( x < π) Fourier ( sin 1x sin 2x g(x) = sin 3x 2 ) +. f f = g Fourier f ( ) c n (f) Fourier F[f](n) (1.23) F[f ](n) = inf[f](n). 35
37 Fourier F [f ] (ξ) = iξf[f](ξ) f k (1.24) F[f (k) ](n) = (in) k F[f](n). (f ) 1 = (Ff ) in 1 Fourier ( ) ( ) f n Fourier Fourier Fourier n ( ) f f f Fourier ( C 1 Fourier ) f : R C C 1 f Fourier f f ( C.4.1, p. 133) Fourier inc n f Fourier c n e inx = c n c n = n 0 n Z n 0 = Weierstrass M test n= n= inc n 2 = f (x) 2 dx. n 2 c n 2 = 1 f (x) 2 dx. ( n c n 1 ) n n 2 c n 2 n 0 n 0 π π f 6 (x) 2 dx <. n= c n e inx f C k 1 n = 2 1 f (x) 2 dx π 2 3 (1.25) n= n 2k c n 2 = 1 f (k) (x) 2 dx n ± n 2k c n 36
38 1.6 Fourier R R 9 X f, g X (f, g) := f(x)g(x) dx (f, g) f g X ( ) f f f := (f, f) {cos mx} m Z 0 {sin nx} n N {e inx } n Z X {φ n } n N (i) ( m, n) m n (φ m, φ n ) = 0 (ii) ( n) (φ n, φ n ) 0 n (iii) ( f X ) lim n f c k φ k = 0, c k := (f, φ k) (φ k, φ k ) k=1 3 f X s N (x) := N k= N c k e ikx, c k = (f, eikx ) (e ikx, e ikx ) = 1 f(x)e ikx dx (n Z) N s N f : lim f s N = 0 f(x) = N n= c n e inx ( ). f Fourier, c n f Fourier f n Fourier F[f](n) F[f ](n) = inf[f](n) (n Z). f Fourier ( ) Fourier ( ) f C 1 Fourier f R lim sup s N (x) f(x) = 0. N x R f [, π] C 1 lim s N(x) = N f(x) f(x + 0) + f(x 0) 2 (x f ) (x f ) Fourier (Gibbs ) 9 [, π] C 1 Lebesgue Lebesgue ( f 2 ) X L 2 (, π) (Hilbert ) 37
39 1.7 : Fourier Fourier Fourier [8] ( ) Lebesgue Hilbert ( ) Fourier L 2 Lebesgue ( Fourier ) Hilbert ( ) ( Lebesgue ( ) Fourier, Lebesgue ( ) Fourier ( ), ) 38
40 2 Fourier 2.0 Fourier ( ) Fourier Fourier 1. Fourier (Fourier ) 2. Fourier ( ) 3. Fourier ( ) 4. Fourier ( ) Fourier f Fourier Ff(ξ) = 1 f(x)e ixξ dx Riemann ( f R ) Ff(ξ) = 1 R2 lim f(x)e ixξ dx R 1,R 2 R 1 ( f ) (Fourier ) x ± f(x) 0 Fourier f Lebesgue ( ) Fourier Lebesgue 2.1 Fourier, Fourier Fourier Fourier (inversion formula) = 4 39
41 f : R C f f(x) x ± (0 ) : l > 0 f [ l, l] Fourier l Fourier l c n := 1 l nπ i f(x)e l x dx (n Z) f(x) = c n e i nπ l x 2l l n= (x ( l, l)). ( : f( l) = f(l) x = ±l ) (c n l ) π l c n := π c n = 1 l nπ i f(x)e l x dx f(x) = l π l n= c ne i nπ l x (x ( l, l)). l x f(x) 0 l l 2 f(ξ) := 1 f(x)e iξx dx (ξ R) c n f ( n π ) (n Z). l ( : c n f ( n π ) 1 nπ = i f(x)e l x dx l 1 x >l n ) f(x) = π l n= 1 c ne i nπ l x 1 π l f (ξ) e iξx dξ n= (x ( l, l)). (2 ξ > 0 F (ξ)dξ ξ ) f(ξ) := 1 x >l f(x) dx 0 n f(nπ/l)e i nπ l x n= f(x)e iξx dx (ξ R) f(x) 1 F (n ξ) f (ξ) e iξx dξ (x ( l, l)). l x R 2 ((i) [ l, l] R (ii) ) f (2.1) f(ξ) := 1 l 2 lim = l l f(x)e iξx dx (ξ R) f(x) = 1 40 f (ξ) e iξx dξ (x R).
42 Fourier f (2.2) f(ξ) := 1 f(x)e ixξ dx (ξ R) f f Fourier (the Fourier transform of f) f f Fourier (Fourier transform, Fourier transformation) F Ff = f g : R C (2.3) g(x) := 1 g(ξ)e ixξ dξ (x R) g g Fourier F F g = g (2.1) Fourier (2.4) F (Ff) = f. (2.5) F (F g) = g. f Fourier Fourier ( g Fourier Fourier g ) Fourier Fourier (Fourier (1)) Fourier (2.6) f(ξ) = 1 f(x)e ixξ dx, g(x) = (2.7) f(ξ) = f(x)e ixξ dx, g(x) = g(ξ)e ixξ dx g(ξ)e ixξ dx ( (2.7) 3 ) 2016 (2.2), (2.3) ( 4 ) 2017 (2.6) ( ) F Ff = f, FF g = g ( ) 3 Wikipedia 4 41
43 2.1.2 (Fourier (2)) Fourier (a) f(ξ) 1 R2 = lim f(x)e ixξ dx R 1,R 2 R 1 ) R2 (b) f L 1 (R) Lebesgue (c) f L 2 (R) f(ξ) = lim ( L 2 φ L 2 = (d) R 1 f(x)e ixξ dx Riemann (Riemann R 1 R R 1/2 φ(x) dx) 2 ) f(x)e ixξ dx ( L 2 Riemann (a) (Lebesgue ) 2016 ( ) (Fourier Fourier ) Fourier Fourier f : R C f Fourier {c n } n Z c n := 1 f(x)e inx dx (n Z) 5 f Fourier ˆf(ξ) = 1 ( ) Fourier f(x) = n= c n e inx (x R) Fourier f(x) = 1 f(x)e iξx dx (ξ R) ˆf(ξ)e iξx dξ (x R) (F 1 F ) F 1 F 6 Fourier Fourier ( ) Lebesgue L 2 (R) F : L 2 (R) L 2 (R) ( unitary ) S ( ) F : S S F 5 f Fourier c n ˆf(n) 6 A = (a ij ) Hermite (a ji ) A 42
44 2.2 Fourier Fourier ( ) Fourier ( Fourier ) ( ) ( ) Mathematica ( ) Fourier ( ) f(x) = e x ( ) Ff(ξ) = (2.8) F [ e x ] (ξ) = Fourier 2 1 π ξ π ξ F[( ) ](Fourier ) (2.8) F[f(x)](ξ) = Ff(ξ). F [ e x ] (y) = 2 1 π y ( Mathematica FourierTransform[] ) 43
45 2.2.1 (1) g = Ff f = F g (2) F f(x) = Ff( x), Ff(ξ) = F f( ξ). f Fourier f Fourier (3) g = Ff Fg(ξ) = f( ξ). (1) F F f = f g = Ff f = F (Ff) = F g. (2) Fourier g Fourier F g(x) = 1 g(ξ)e iξx dξ (x R) F g f Fourier (2.9) F f(x) = 1 f(ξ)e iξx dξ (x R) F f Fourier f Fourier Ff(ξ) = 1 f(x)e iξx dx (ξ R) Ff f Fourier (2.10) Ff(x) = 1 f(ξ)e iξx dξ (x R) Ff (2.9) (2.10) F f(x) = Ff( x) (x R) Ff(ξ) = Ff( ( ξ)) = F f( ξ) (ξ R). (3) (1), (2) Ff(ξ) = F f( ξ) (a) I = 1 0 (2 + 3x + 4x 2 )dx = 1 0 (2 + 3y + 4y 2 )dy. (b) f f(x) = 1 + 2x (x R) f(y) = 1 + 2y (y R) (c) D Df = f Dg = g 44
46 ( ) ( Fourier ) a > 0 (1) F [ e a x ] 2 a (ξ) = π ξ 2 + a. 2 [ (2) F 1 x 2 + a 2 (3) f(x) := ] (ξ) = 1 a π 2 e a ξ. 1 ( a < x < a) 2a 0 ( ) Ff(ξ) = 1 sin(aξ). aξ sinc x := sin x x Ff(ξ) = 1 sinc(aξ). [ ] sin (ax) (4) F (ξ) = ax [ (5) F e ax2] (ξ) = 1 e ξ2 4a. 2a 1 2a ( ξ < a) 0 ( ξ > a) 1 4a (ξ = ±a).. ( ) (1) 1 R 0 e a x e ixξ dx = 1 R 0 = 1 1 e (a+iξ)r a + iξ e (a+iξ)x dx = 1 [ ] e (a+iξ)x R (a + iξ) a + iξ ( e ( a+iξ)r = e ar 0 (R ) 7 ) (R ). x = y ( ) 1 0 e a x e ixξ dx = 1 R R 0 1 e ay e iyξ dy 1 1 a iξ e a x e ixξ dx = 1 ( 1 a + iξ + 1 ) = 1 a iξ (2.11) F [ e a x ] (ξ) = 2 a π ξ 2 + a. 2 (R ). 2a 2 ξ 2 + a = 2 π 2a ξ 2 + a 2. 7 z = x + iy (x, y R) e z = e x+iy = e x (cos y + i sin y) = (e x cos y) 2 + (e x sin y) 2 = e x. e z = e Re z. 45
47 (2) [ F 1 x 2 + a 2 ] (ξ) = 1 e ixξ x 2 + a 2 dx. ( 2.2.5) (1) Ff = g Fg(ξ) = f( ξ) (2.11) [ ] 2 a F (ξ) = e a ξ = e a ξ. π x 2 + a 2 [ F 1 x 2 + a 2 ] (ξ) = 1 π a 2 e a ξ. (3) Ff(ξ) = 1 = [ 1 F 1 2a sin(ax) ax f(x)e ixξ dx = 1 a e iaξ e iaξ iξ ] (ξ) = = 1 aξ a 1 2a e ixξ dx = e iaξ e iaξ 2i [ ] 1 e ixξ x=a 2a iξ x= a 1 = aξ sin(aξ) = 1 sin(aξ). aξ (4) Ff = g Fg(ξ) = f( ξ) f : f( ξ) (ξ f ) f( ξ + 0) + f( ξ 0) 2 (ξ f ). ( (Fourier Fourier ) Fourier Fourier 8 ) f( ξ) = f(ξ) [ ] sin(ax) F (ξ) = ax 1 2a ( ξ < a) 0 ( ξ > a) 1 4a ( ξ = a). (5) (2.2.4) (sinc ) sinc ([sínk], the sinc function, the cardinal sine function) Woodward- Davies [10] (1952 ) (2.12) sinc x := sin(πx) πx (Mathematica ) sinc x := sin x (2.12) the normalized sinc function ( x ) 8 [9] 46
48 [10] Sampling analysis rests on a well-known mathematical theorem that if a function of time f(t) contains no frequencies greater than W, then f(t) r f(r/2w ) sinc(2w t r)... (24) where sinc x is an abbreviation for the function (sin πx)/πx. This function occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own. sinc Its most important properties are that it is zero when x is a whole number but unity when x is zero, and that and sinc x dx = 1 sinc(x r) sinc(x s) dx = { 1, r = s) 0, r s r and s both being integers. abbreviation ( ) whole number x = 0 sinc x = 1 x sinc x = 0 sin x dx = π sinc x dx = 1 x π sign (a ξ) + sign (ξ + a) Mathematica (4) α < β 2 2 α, β sign (ξ α) + sign (β ξ) 2 = α = a, β = a 0 (ξ < α ξ > β) 1 (α < ξ < β) 1 2 (ξ = α, β). sinc (Stenger [11] ) e ax2 Fourier e ax2 (a ) [ (2.13) F e ax2] (ξ) = 1 e ax2 e iξx dx = 1 e ξ2 4a 2a Fourier ( ) a = 1/2 : [ ] F e x2 /2 (ξ) = e ξ2 /2. 47
49 f(x) := e x2 /2 Ff = f f Fourier F 1 f 0, 1 9 ( ) 1 e x2 /2 1 ( ) 2 ax 2 iξx = a (x 2 + iξa ) x F [e ax2] (ξ) = e ξ2 4a 1 ( = a x + iξ ) 2 ξ2 2a 4a iξ a(x+ e 2a) 2 dx = e ξ2 4a 1 e ax2 dx. Cauchy ( ) ( ) e x2 dx = π ( ) ax = y e ax2 dx = e 1 π y2 dy =. a a (2.14) F ( ) 4 X, X, X + iξ 2a Cauchy 0 = = = C e az2 dz e az2 dz + [ X,X] X X e ax2 dx + [ e ax2] (ξ) = 1 2a e ξ2 4a. iξ a(x+ e 2a) 2 dx = e ax2 dx [X,X+i 2a] e az2 ξ [X,X+i 2a] e az2 ξ, X + iξ 2a dz + dz X X, X C ( ) [X+i ξ 2a, X+i ξ 2a iξ a(x+ e 2a) 2 dx + e az2dz +,] e az2 [ X+i ξ, X] 2a X 2, 4 0 z = x + iy (x, y R) e az2 = e Re( az 2 ) = e a(x 2 y 2). [ ] X, X + i ξ 2a x = X, y ξ e az2 ξ e 2 4a e ax 2 (z [X,X+i 2a] e az2 ξ dz [X,X+i 2a] ξ 2a e az2 ξ dz e 2 9 m, σ 2 1 σ exp 48 [ X, X + i ξ 2a ] ). 4a e ax 2 ξ 2a 0 e az2 [ X+i ξ, X] 2a dz. ( X ). ( (x m)2 2σ 2 ) m = 0, σ = 1 dz
50 ( ) 10 g(ξ) := 1 g (ξ) = 1 ( ix)e ax2 e ixξ dx = 1 ( ) i e ixξ dx 2a e ax2 = 1 ([ i 2a e ax2e ixξ] = ξ 2a = ξ 2a g(ξ). 1 ( ) ξ = 0 g(0) = 1 e ax2 e ixξ dx Y = g(ξ) e ax2 e ixξ dx ) i ( iξ) e ixξ dx 2a e ax2 e ax2 dx = 1 π = 1. a 2a dy dξ = ξ 1 Y, Y (0) = 2a 2a ( : dy dξ = ξ dy 2a Y Y Y = 1 e ξ2 2a 4a.) = g(ξ) = 1 2a e ξ2 4a. ξ ξ2 dξ. C log Y = + C. 2a 4a : 1 x 2 Fourier + a ( ) P (z), Q(z) C[z], deg P (z) deg Q(z) + 1, x R P (x) 0, p > 0 f(z) := Q(z) P (z) f(x)e ipx dx = i Res ( f(z)e ipz ; c ). Im c>0 (f 10 [12] 49
51 1 0 p > 0 ) (2.15) e iξx πe a ξ dx = x 2 + a2 a (ξ R) ( ξ ξ = 0 ) (2.15) ξ > 0, ξ = 0, ξ < 0 ξ < 0 ξ > ( ) e iξx e i( ξ)z dx = i Res x 2 + a2 z 2 + a ; ia = i e iξz 2 z + ia = πeaξ z=ia a. ξ = 0 tan 1 x π a x = y e ixξ x 2 + a 2 dx = e iξy πea( ξ) dy = y 2 + a2 a ξ > 0 = πe aξ a. πe a ξ a 1 [ F 1 x 2 + a 2 e iξx x 2 + a dx = 1 π 2 a 2 e a ξ. ] (ξ) = 1 π a 2 e a ξ Mathematica Mathematica FourierTransform[] InverseFourierTransform[] F[f] Fourier ( ) Mathematica ( N ) Mathematica Fourier FourierParameters->{0,-1} 11 f[x] Fourier ( y ) 11 Mathematica 1 f(x)e ixξ dx ixy ixy FourierParameters->{0,-1} 50
52 F[f(x)](y) FourierTransform[f[x],x,y,FourierParameters->{0,-1}] { } 1 ( 5 < x < 5) 2 sin 5ξ f(x) = 0 ( ) π ξ Mathematica f[x_]:=if[-5<x<5,1,0] FourierTransform[f[x],x,y,FourierParameters->{0,-1}] Fourier,FourierParameters->{0,-1}] myf[fx_,x_,y_]:=fouriertransform[fx,x,y,fourierparameters->{0,-1}] myf[f[x],x,y] 2.3 Fourier ( ) Paresval Fourier (2.16) (2.17) F(f 1 + f 2 ) = Ff 1 + Ff 2, F(λf) = λff Fourier Fourier (2.18) Ff(ξ) = F f( ξ). F g(x) = 1 g(ξ)e ixξ dξ 12 Fourier Fourier 51
53 F f(ξ) = 1 F f( ξ) = 1 f(x)e iξx dx. f(x)e iξx dx = Ff(ξ) (2.19) (2.20) F[f(x a)](ξ) = e iaξ Ff(ξ), F [ f(x)e iax] (ξ) = Ff(ξ a). x a = y (, ) (, ) x = y + a, dx = dy F [f(x a)] (ξ) = 1 = e iaξ 1 f(x a)e ixξ dx = 1 f(y)e iyξ dy = e iaξ Ff(ξ). f(y)e i(y+a)ξ dy a 0 (2.21) F[f(ax)](ξ) = 1 a Ff ( ) ξ. a y = ax (, ) (, ) dy = a dx, x = y a F [f(ax)] (ξ) = 1 = 1 1 a f(ax)e ixξ dx = 1 f(y)e iy(ξ/a) dy = 1 a Ff f(y)e i y a ξ 1 a dy ( ) ξ. a ( a ) Fourier (2.22) F [f (x)] (ξ) = (iξ)ff(ξ). ( k N) F [ f (k) (x) ] (ξ) = (iξ) k Ff(ξ). 52
54 f(x) 0 ( x ± ) F [f (x)] (ξ) = 1 f (x)e ixξ dx = lim R 1 R R f (x)e ixξ dx ( 1 [f(x)e = lim ] R ) ixξ x=r f(x)( iξ)e ixξ dx R x= R R = 1 ) (0 f(x)( iξ)e ixξ dx 1 = iξ f(x)e ixξ dx = iξff(ξ) Fourier (2.23) ( k N) d dξ Ff(ξ) = d dξ = 1 d Ff(ξ) = if [xf(x)] (ξ). dξ 1 ( ) k d Ff(ξ) = ( i) k F [ x k f(x) ] (ξ). dξ f(x)e ixξ dx = 1 ( ix)f(x)e ixξ dx 1 = i xf(x)e ixξ dx = if [xf(x)] (ξ). ( ) f(x)e ixξ dx ξ 2.4 ( ) Fourier Fourier Fourier Fourier 2.5 : ( ) ( ) Fourier Fourier ( ) f Ff f Ff 53
55 (1) f R Ff Ff f 1. f = sup f(x), f 1 = x R f(x) dx. (2) f xf(x) R Ff C 1 d Ff(ξ) = F[( ix)f(x)](ξ). dξ ( ) k N f x k f(x) R Ff C k ( ) k d Ff(ξ) = F [ ( ix) k f(x) ] (ξ) (ξ ± ). dξ ( ) 1 (3) f f lim f(x) = 0 F[f ](ξ) = iξff(ξ), Ff(ξ) = O x ± ξ (ξ ± ). ( ) k N f C k f (j) (j = 0, 1,..., k) R F [ ( ) f (k)] (ξ) = (iξ) k 1 Ff(ξ), Ff (ξ) = O (ξ ± ). ξ k (2), (3) ( ) k d F = F [ ( ix) k ] ( ) k d, F = (iξ) k F. dx dx x ± f(x) Ff f ξ ± Ff(ξ) Fourier [13] (Riemann ) [14] (Lebesgue ) (2 ) (i) f(x) := e a x x = 0 ( C 1 ) x ± f(x) 14 1 Ff(ξ) = ξ ± ( Ff ξ 2 + a2 ξff(ξ) 15 ) Ff C 1 ( x < a) (ii) f(x) = 2a (, C 1 0 ( x > a) ) x ± ( 0 ) Ff(ξ) = sin(aξ) ξ ± ( 0 Ff ξ 16 ) F(ξ) C ξff(ξ) 1 16 sin(aξ) ξ ξ sin(aξ) ξ ( ) 54
56 (iii) f(x) = e ax2 C x ± Ff(ξ) = e ξ2 4a ξ ± Ff C ( ) : 55
57 3 Fourier Fourier ( ( ) Fourier ) Fourier Fourier (Fast Fourier Transform, FFT) Fourier Fourier Fourier f ( ) Fourier N N N T {f j } Fourier c n = 1 f(x)e in π T x dx C n T 0 {C n } N C N f 0 f 1. C 0 C 1. CN f C Fourier W = 1 1 ω 1 ω 1 2 ω 1 () N 1 ω 2 1 ω 2 2 ω 2 ()... 1 ω () 1 ω () 2 ω ()() (W (n, j) 1 N ω (n 1)(j 1) ) ω = e i/n ω 1 ω 1 2 ω 1 () W 1 = 1 ω 2 1 ω 2 2 ω 2 ()... 1 ω () 1 ω () 2 ω ()() (W 1 (j, n) ω (j 1)(n 1) ) U := NW unitary ( Fourier ( ) unitary ) 56
58 3.1 Fourier 1 Fourier T f : R C T ( ) (3.1) c n := 1 T T 0 in f(x)e T x dx (n Z) ( [ T/2, T/2] [0, T ] ) (3.2) f(x) = n= in c n e T x (x R). 1 [T ] N (3.3) h := T N, x j = jh (j Z) x j (3.4) f j := f (x j ) (j Z) 1 {f j } j=0 1 {c n } {C n } ( ) h (, sampling period), 1 (, sampling rate, sample h rate) {f j } ( ) ( ) ( ) Fourier 2 ( ( ) ) Fourier ( ) I = b a F (t) dt {t j } N j=0 [a, b] N I I N := N j=1 ( F (t j 1 ) + F (t j ) F (t0 ) h = h + F (t 1 ) + + F (t ) + F (t ) N), h := b a N 1 f {f j } n Z f j+n = f j (j Z) N {f j } j Z N {f j } j=0 2 57
59 F b a F (t 0 ) = F (a) = F (b) = F (t N ) : (3.5) I N = h F (t j ) j=0 ( ). (3.1) C n : C n := 1 T h (3.6) ω := e i T h = e i/n in e T x j (3.7) C n = 1 N j=0 in f(x j )e T x j. ( T h = T T N = N ) in = e T jh = ω nj C n = 1 T T f j ω nj. N j=0 j=0 f j ω nj. f Fourier (1 N ) N N ω = e i/n (1), (2) (1) ω 1 N (i) 1 m N 1 ω m 1 (ii) ω N = 1 (2) m Z { j=0 ω mj = N (m 0 (mod N)) 0 ( ). (1) m Z ω m = e i N m m 0 (mod N) ω m = 1 1 m N 1 m ω m 1, ω N = 1 58
60 (2) m 0 (mod N) ω m = 1 j ω mj = 1. ω mj = j=0 1 = N. ω m 1 j=0 ω mj = j=0 (ω m ) j = 1 (ωm ) N 1 ω = 1 ( ) ω N m m 1 ω = 0. m j=0 2 l, m l m (mod N) l m PDF = ( ) ( Fourier ) T f : R C N N h := T N, ω := ei T h = e i/n, x j := jh, f j := f (x j ) (j Z), C n := 1 N N=1 j=0 f j ω nj (n Z) Fourier {C n } n Z (1), (2) (1) {C n } n Z N : C n+n = C n (n Z). (2) n= c n < n Z (3.8) C n = m n c m. m n m n (mod N) m Z (1) ω (n+n)j = ω nj ω Nj = ω nj (2) f(x) = n= C n+n = 1 N in c n e T x f j = f(x j ) = n= j=0 f j ω (n+n)j = 1 N in c n e T x j = n= j=0 f j ω nj = c n. inj c n e T h = n= c n ω nj. 59
61 C n = 1 N = 1 N j=0 m= f j ω nj = 1 N c m j=0 j=0 ( ω nj m= c m ω mj ) ω (m n)j = 1 c m N = c m. N m n m n (1) {C n } n Z N {C n } n=0 ( ) C 0 = m 0 c m = c 0 + c N + c N + c 2N + c 2N +, C 1 = m 1 c m = c 1 + c 1 N + c 1+N + c 1 2N + c 1+2N +, C 1 = m 1 c m = c 1 + c 1+N + c 1 N + c 1+2N + c 1 2N +, C 2 = m 2 c m = c 2 + c 2 N + c 2+N + c 2 2N + c 2+2N +, C 2 =. m 2 c m = c 2 + c 2+N + c 2 N + c 2+2N + c 2 2N +, C n = c n + (c n+pn + c n pn ). p=1 Q C n c n ( c n C n C n c n ) A Yes n Z lim C n = c n. N : h, x j, f j, ω, C n N h N, x j,n, f j,n, ω N, C n,n lim C n,n = c n ε-n N ( n N)( ε > 0)( n N)( N N : N n ) C n,n c n < ε. Q C 1,N = C,N = c 1 + c 1+N + c 1 N + c 1+2N + c 1 2N + c 1, c, A C c 1 c lim (c 1+N + c 1 N + c 1+2N + c 1 2N + ) = 0 N lim N C 1,N = c 1. C n ( n N/2) c n n N/2 60
62 3.2 Fourier N f = C 0 C 1. C f 0 f 1. f CN Fourier C n = 1 N j=0 ω nj f j C = CN f Fourier C N f C C N Fourier N 1 ( 1 ) 0 i (i, j) i i n ( Fourier ) N N ω := e i/n, ω 0 ω 0 ω 0 ω 0 W := 1 ω 0 ω 1 ω 2 ω () N ω 0 ω 2 ω 4 ω 2(), ω 0 ω () ω ()2 ω ()() f = f 0 f 1., C = C 0 C 1. f C (3.9) C n = 1 N j=0 f j ω nj (n = 0, 1,, N 1) C = W f ( ). W ω 0 ω 0 ω 0 ω 0 ω 0 ω 1 ω 2 ω W 1 = ω 0 ω 2 ω 4 ω 2() ω 0 ω ω ()2 ω ()() C n = 1 N j=0 f j ω nj (n = 0, 1,, N 1) f j = 61 n=0 ω jn C n (j = 0,..., N 1).
63 ( 0 W (n, j) 1 N ω nj, W 1 (j, n) ω jn ) (3.9) 0 (N 1 ) W (n, j) 1 N ω nj W (j, k) ω jk (ω jk ) (n, k) { { 1 N ω nj ω jk = 1 ω (k n)j = 1 N (k n 0) 1 (k = n) = = δ kn. N N 0 ( ) 0 ( ) k=0 k=0 W 1 = ( ω jk) f f j = f j = c m ω mj = n=0 m n c m ω nj = n=0 m n n=0 ω nj m n c m = n=0 n=0 c n ω nj ω nj C n W 1 ω jn W f (W unitary ) U := NW (3.10) U = 1 N ( ω nj ), U 1 = 1 N W 1 = 1 N ( ω nj ) ω = ω 1 U Hermite U U = 1 N ( ω jn ) = 1 N ( ω nj ) = U 1. U unitary ( ) Fourier c n = 1 f(x)e inx dx, f(x) = 0 c n = 1 0 n= c n e inx f(x)e inx dx, f(x) = 1 c n e inx { } 1 ( e inx ) Fourier U W U := NW = 1 N (ω nj ) unitary 62 n Z
64 3.2.5 ( Fourier ) Fourier c n = 1 f(x)e inx dx, f(x) = c n e inx n= Fourier (Fourier Fourier ) f(ξ) = 1 f(x)e ixξ dx, f(x) = 1 f(ξ)e ixξ dξ Fourier ( Fourier Fourier ) C n = 1 N j=0 f j ω nj, f j = n=0 C n ω nj Fourier (f {c n } {f j } {C n } ) Fourier Fourier Fourier L 2 (R) unitary ( ) f := f 0 f 1 f 2. f, φ n := f = n=0 c n φ n ω n 0 ω n 1 ω n 2. ω n () C N {φ n } f (φ n, φ m ) = j=0 ω nj ω mj = j=0 ω nj ω mj = j=0 ω (n m)j = { N (n = m) 0 (n m) ( ) {φ n } f = n=0 C nφ n C n = (f, φ n) (φ n, φ n ) = j=0 f jω nj N = 1 N j=0 f j ω nj. 63
65 3.2.6 ( Fourier ) T u: R C Fourier M in u(t) = c n e T t n= M u Fourier {c n } n > M c n = 0 N > 2M N N Fourier {C n } n=0 ( ) C n = c n (0 n M), C N n = c n (1 n M), C n = 0 (M < n < N M) ( (0 ) Fourier {c n } M n= M Fourier {C n} n=0 ) ( ) M = 1, N = 10 C 0 = c 0 + c 10 + c 20 + c 20 + c 30 + = c = c 0, C 1 = c 1 + c 9 + c 11 + c 19 + c 21 + = c = c 1, C 9 = c 9 + c 1 + c 19 + c 11 + c 29 + c 21 + = 0 + c = c 1, C 2 = c 2 + c 8 + c 12 + c 18 + = = 0, C 8 = c 8 + c 2 + c 18 + c 12 + = = 0, 2 n 8 C n = 0. M = 5, N = 10 C 0 = c 0 + c 10 + c 20 + c 20 + c 30 + = c = c 0, C 1 = c 1 + c 9 + c 11 + c 19 + c 21 + = c = c 1, C 9 = c 9 + c 1 + c 19 + c 11 + c 29 + c 21 + = 0 + c = c 1, C 2 = c 2 + c 8 + c 12 + c 18 + = = c 2, C 8 = c 8 + c 2 + c 18 + c 12 + = = c 2,.. C 4 = c 4 + c 6 + c 14 + c 16 + = c = c 4, C 6 = c 6 + c 4 + c 16 + c 14 + = 0 + c = c 4, C 5 = c 5 + c 5 + c 15 + c 15 + = c 5 + c = c 5 + c 5. C 5 = c 5 C 5 = c 5 M = 5 N > 10 (( ) ) N > 2M ( ) 64
66 0 n M C n = m n c m = c n + (c n+pn + c n pn ). p=1 n + pn N > 2M > M c n+pn = 0. n pn M N < M c n pn = 0. C n = c n. 3.3 : (2017/11/23 ) W W ( ) in φ n (x) = e T x (1 ) φ n = (ω n 0, ω n 1,, ω n() ) T (n = 0, 1,, N 1) C N 3 N ( 1 N ) 0 N 1 : x = x 0 x 1.. : A = a 00 a 01 a 0, a 10 a 11 a 1,..... x a,0 a,1 a, (i, j) a ij A = (a ij ) i i (n, j) : A = (a nj ). T ( t ) (Hermite ) C N x, y (x, y) (x, y) := (a nj ) T = (a jn ), (a nj ) = (a jn ). j=0 (3.11) (x, y) = (y 0 y 1 y ) 3 ω ω = e i N k=0 x j y j x 0 x 1. x = y x ω pk p 0 (mod N) N, 0 65
67 ( ) T (> 0) f : R C N N h := T N, x j := jh (j Z), f := f(x 0 ) f(x 1 ). f(x ) f C N ( h ) in (1 ) φ n (x) := e T x (3.12) (φ n, φ m ) = T 0 T in e T x im e T x i(n m) dx = e T x dx = T e i(n m)θ dθ = T δ nm 0 0 {φ n } n Z ( ) φ n φ n j ω nj : (3.13) φ n = ( ω n 0, ω n 1,, ω n()) T. φ n = in e T x 0. in e T x j. in e T x = ω n 0. ω n j. ω n () ( in T x j = in T j T N = nj i N ). φ n+n = φ n ( ) 0 n N 1, 0 m N 1 n, m (3.14) (φ n, φ m ) = Nδ nm (φ n, φ m ) = k=0 ω nk ω mk = k=0 ω nk ω mk = k=0 ω k(n m) = Nδ nm. φ 0,, φ C N ( ) Φ : ω 0 0 ω 0 1 ω 0 2 ω 0 () ω 1 0 ω 1 1 ω 1 2 ω 1 () (3.15) Φ := (φ 0 φ 1 φ ) = ω 2 0 ω 2 1 ω 2 2 ω 2 ()..... ω () 0 ω () 1 ω () 2 ω () () 66
68 Φ (n, j) ω nj Φ Φ Φ Φ = φ 0 φ 1. ) (φ 0 φ 1 φ = φ 0φ 0 φ 0φ 1 φ 0φ φ 1φ 0 φ 1φ 1 φ 1φ... φ = (φ nφ j ) = (Nδ nj ) = NI. φ φ 0 φ φ 1 φ φ (3.16) Φ 1 = 1 N Φ = 1 N (ω nj) = 1 N (ω jn) = 1 N ( ω jn ). 11/22 W Φ 1 ( ) W = 1 ( ) ω jn = Φ 1. N W 1 = Φ = ( ω nj). 3.4 FFT (the fast Fourier transform) FFT ( [15], [16] ) FFT Fourier N N 2 N = 2 m (m ) ( ) O(N log N) Fourier FFT N CD 44.1 khz = ( ) = ( ) Fourier Fourier Fourier Fourier 67
69 3.5 Fourier ( 2016 ) {C n } f : R C N Fourier (3.17) h = N, x j = jh, f j = f(x j ), ω = e ih, C n = 1 f j ω nj. N j=0 N S N (x) (3.18) S N (x) := N 2 1 k= N 2 2 k= 2 C k e ikx C k e ikx (N ) (N ) S N (x j ) = f(x j ) (j Z) s N f {S N (x j )} {C n } Fourier (3.18) Fourier n= n= c n e inx (c n := 1 f(x)e inx dx) 0 c n e inx = lim n s n (x), s n (x) := s n (x) n k= n c k e ikx s n (x) = a n (a k cos(kx) + b k sin(kx)), a k = 1 π 0 k=1 f(x) cos(kx)dx, b k = 1 π 0 f(x) sin(kx)dx Fourier C k Fourier ( ) s n (x) Fourier c k C k n S n (x) = C k e ikx k= n ( ) 2n + 1 Fourier C k ( k n) N 2n + 1 N = 2n + 1 N N 68
70 ( N = 2 m ) (3.19) S N (x) := Ck e ikx, k N 2 k N/2 (a) N k N/2 k k =, 2 1,..., ( N ) 2 Ck e ikx := k N 2 k N/2 C k e ikx = k= 2 k= 2 C k e ikx. (b) N N/2 k N/2 k k = N/2, N/2+1,, N/2 N + 1 (1 ) 4 Ck e ikx := k N 2 N/2 1 k= N/2 C k e ikx x = x j = jh e ikx = e ikx j = e ikjh = ω kj k N C k e ikx = C k ω kj k N S N (x j ) = N/2 1 k= N/2 ()/2 C k ω kj k= ()/2 C k ω kj (N ) (N ) = k=0 C k ω kj = f j = f(x j ) S N (x) = N/2 k= N/2+1 C k e ikx S N (x j ) = k=0 C k ω kj = f j = f(x j ) x j x R S N (x) 1 x S N (x) (3.18) x j+1/2 = x j + h/2 (j = 0, 1,, N 1) (Fourier (2.20) ) 4 k N 2 Ck e ikx := N/2 k= N/2+1 C k e ikx 2 69
71 x = x j + x = jh + x S N (x) = Ck e ikx = Ck e ik(jh+ x) = ( Ck e ik x) ω kj k N 2 k N 2 k N 2 Fourier C k e ik x Fourier S N (x) FFTPACK C ( FFTW ) // workn[] zffti(n, workn); // c[k] (0 k N-1) C_k // x=j h+dx (j=0,1,...,n-1) f_n(x) for (k = 0; k < N/2; k++) { d[k] = CMPLX(cos(k*dx),sin(k*dx)); d[n-k] = conj(d[k]); } // N N if (N % 2 == 0) d[n/2] = CMPLX(cos(N*dx/2),-sin(N*dx/2)); for (k = 0; k < N; k++) r[k] = c[k] * d[k]; zfftb(n, r, workn); // r[j] f_n(x_j+dx) 3.6 Mathematica Fourier Mathematica Fourier[f ] f = {f 0, f 1,..., f } 1 f n ω nj (j = 0, 1,, N 1) N n=0 ( ) ( Mathematica Fourier ) InverseFourier[C ] Fourier ( N ) Fourier Fourier[f,FourierParameters->{-1,-1}] N C = (c ij ) (circulant) N L 0, L 1,, L c ij = L l, l = (j i) mod N L 0 L 1 L N 2 L L L 0 L 1 L N 2 C = L 2... L 0 L 1 L 1 L 2 L L 0 70
72 U := 1 N (ω nj ) φ 0 0 U φ 1 CU = diag [φ 0, φ 1,, φ ] =..., φ p := 0 φ j=0 ω pj L j C C 1, det C ( ) 3.8 : ( ) Fourier cos, sin : Fourier C n, A n, B n f : R C c n ( N ) C n a n, b n A n, B n : (3.20) A n := 2 N j=0 f j cos(nx j ), B n := 2 N j=0 f j sin(nx j ). h := N, x j = jh, f j = f(x j ). a n, b n, c n ( k N) C k = (A k ib k ) /2, C k = (A k + ib k ) /2, C 0 = A 0 /2, ( k N) A k = C k + C k, ib k = C k C k, A 0 = 2C 0, A 0 n n ( n N) 2 + (A k cos kx + B k sin kx) = C k e ikx. {A n }, {B n } N f B k = 0 C k = C k, A k = 2C k. f A k = 0 C k = C k, B k = 2iC k. k=1 k= n Fourier f : R R f A k, B k R N Fourier C N k = C k = C k A N k = A k, B N k = B k. 71
73 N A 0, A 1, B 1, A 2, B 2,, A N/2 1, B N/2 1, A N/2, N A 0, A 1, B 1, A 2, B 2,, A ()/2, B ()/2 (N f B N/2 = 0 ) N Fourier N R N (f 0, f 1,, f ) (A 0, A 1, B 1,, A N 1, B N 1, A N ), A n = 2 N B n = 2 N j=1 j=1 f j cos nj N f j sin nj N (n = 0, 1,, N 2 ), (n = 1, 2,, N 2 1). f j = A N/2 1 n=1 ( A n cos nj N + B n sin nj ) + A N/2 ( 1) j (j = 0, 1,, N 1). N N ( R N (f 0, f 1,, f ) A 0, A 1, B 1,, A 2, B 2 ), A n = 2 N B n = 2 N j=0 j=0 f j cos nj N f j sin nj N ( n = 0, 1,, N 1 ), 2 ( n = 1, 2,, N 1 ). 2 f j = A ()/2 n=1 ( A n cos nj N + B n sin nj ) N (j = 0, 1,, N 1). (C N int N N/2 2 N 0 B N/2 N for (n=0;n<=n/2;n++) for (n=1;n<=n/2;n++) ) (f 0, f 1,, f N ) R N+1 ( (3.21) A n = 1 f f j cos πnj N N j=1 + ( 1)n f N ) (n = 0, 1,, N) 72
74 (f 0, f 1,, f N ) (A 0, A 1,, A N ) R N+1 R N+1 ( ) (3.22) f j = 1 A A n cos πnj 2 N + ( 1)j A N n=1 (j = 0, 1,..., N). (3.22) Fourier A n f [, π] R 2N Fourier C n (2N) A n = 2C (2N) n (n = 0, 1,, N) f : [0, π] R R f 2N Fourier 2N Fourier C n, A n, B n h = 2N = π N, x j = jh, f j = f(x j ) B n = 0, A n = 2C n, A n+2n = A n, A 2N n = A n A 0, A 1,, A N ω := exp i f 2N 2N j = f j = f j, ω n(2n j) = ω nj A n = 2C n = 1 N ( = 1 f 0 + N ( = 1 f 0 + N ( = 1 N f j=0 j=1 j= f j = 2 j=0 j=1 C n ω nj = C 0 + f j ω nj f j ω nj + f N ω nn + j=1 f j ( ω nj + ω nj) + ( 1) n f N ) f j cos πnj N n=1 + ( 1)n f N C n ω nj + C N + ) f 2N j ω n(2n j) ). n=1 ( ) = 1 ( A 0 + An ω nj + A n ω nj) + ( 1) j A N 2 n=1 ( ) = 1 A A n cos πnj 2 N + ( 1)j A N. n=1 73 C 2N n ω (2N n)j
75 3.8.4 (f 1,, f ) R (3.23) B n = 2 N j=1 f j sin πnj N (n = 1, 2,, N 1) (f 1, f 2,, f ) (B 1, B 2,, B ) R R (3.24) f j = n=1 B n sin πnj N (j = 1, 2,, N 1). B n f [, π] R 2N Fourier C n (2N) B n = 2iC n f(0) = f(π) = 0 f : [0, π] R R f 2N Fourier h = 2N = π N, x j = jh, f j = f(x j ) 2N Fourier C n, A n, B n A n = 0, B n = 2iC n, B n+2n = B n, B 2N n = B n C 0 = A 0 2 B 0 = 0, B 2N N = B N B N = 0 B 1, B 2,, B f 0 = f N = 0, f 2N j = f j = f j, ω n(2n j) = ω nj B n = 2iC n = i N = i N = i N = 2 N ( j=1 j=1 j=1 2 j=0 f j ω nj + f j ω nj j=1 f j ( ω nj ω nj) f j sin πnj N. f 2N j ω n(2n j) ) 74
76 3.2.1 f j = = 1 2i 2 j=0 n=1 C n ω nj = C 0 + n=1 ( Bn ω nj B n ω nj) = B n sin πnj N. n=1 C n ω nj + C N + n=1 C 2N n ω (2N n)j 75
77 4 ( Mathematica ) (1) WWW 1 WAVE guitar-5-3.wav 2 ( ) 3 (2) Mathematica (1) ( ) SetDirectory["~/Desktop"] FileNames[] guitar-5-3.wav Fourier ( [17]) snd=import["guitar-5-3.wav","sound"] guitar-5-3.wav snd (snd sound ) tbl = snd[[1, 1, 1]]; tbl (tbl table (, ) ) tbl=snd[[1,1,2]] {ltbl,rtbl}=snd[[1,1]] snd[[1,2]] sr=snd[[1,2]] (sr Sample Rate ) CD 44.1 khz snd = Import[" URL Import 3 Safari control 76
78 tb = Take[tbl, {1, 3*sr}]; g = ListPlot[tb, PlotRange -> All] sr 3 ( sr khz 1 3 sr 3 Take[] ) tb = Take[tbl, { , sr}]; g = ListPlot[tb, Joined -> True, PlotRange -> {{1, 1600}, {-0.3, 0.3}}] (sr 1 s = ) 1600 (1600/ ) ListPlay[tb, SampleRate->sr] tb c = Fourier[tb]; ListPlot[Abs[c], Joined->True, PlotRange->All] tb Fourier c ( ) Abs[] Re[], Im[] ( C n = C N n ) (* n1 n2 c[[n]] *) graph[c_, n1_, n2_] := ListPlot[Abs[c], Joined -> True, PlotRange -> {{n1, n2}, {0, Max[Abs[c]]}}] graph[c, 1, 1600] graph[c, 120, 140] ( ) 130 C 129 ( c 1 c[[1]] C 0 Fourier 1 ) 129 Hz ( 131 Hz ( ) ) ( ) Fourier tb2=inversefourier[c]; Norm[tb-tb2] tb2=re[inversefourier[c]]; ( ) 77
79 4.2 PCM ( ) PCM (pulse code modulation, ) ( ) ( ) 1 CD,, (a) ( ) (b) ( ) LPCM (linear PCM) ( ) ( AD (analog-to-digital conversion) ) 1 CD (1980 SONY Phillips 4 ) 44.1 khz 44.1kHz (a) 20 Hz 20 khz (b) 2 ( ( 5.0.2) ) 2 20 khz = 40 khz 2 ( ) = = CD (CD ) CD (2 ch) k 16 b 2 = kb = kb = KB MB ( 1 MB = 1024 KB ) MB CD CPU 1.2 MB 78
80 CD ( MB ) 2016 CD MP ( x ) t x x(t) T x: R C Fourier (4.1) (4.2) x(t) = c n = 1 T n= T 0 c n e i nt T (t R), t i x(t)e T dt (n Z). f = 1 T n T n, n f. n 0 n 0 f n = ±n 0 [0, T ] N ( ) T s = T/N, f s = N T [0, T ] N t j := jt s x x j = x(t j ) Fourier C n (4.3) C n = 1 N j=0 x j ω nj (n Z), ω = e i/n. {C n } N {C n } n=0 {C n } n=0 C n = {C n } {x j } Fourier : (4.4) x j = n=0 p n c p C n ω jn (j = 0, 1,..., N 1) fs = 44.1 khz T = 1 s (N = f s T = ) T = 1 s Fourier 79
81 4.3.3 C n (1 n N 1) u c n = c n c n = c n. c n = 1 T T 0 in x(t)e T t dt = 1 T T 0 in x(t) e T t dt = 1 T T 0 i( n) x(t)e T t dt = c n. (Cf. f Fourier f f(ξ) = f( ξ) ) C n = C n = C N n, C n = C n = C N n n (1 n N), C n ( Fourier[] ( Fourier ) ) n n /T (4.1) n T n, n T ) c 1, c 1 1 Hz c 2, c 2 2 Hz = n Hz (T = 1 s.. C 129 = C Hz C 258 = C N Hz 129 Hz (1 ) L ( ) 1 c u tt(x, t) = u 2 xx (x, t) (0 < x < L, t > 0) u(0, t) = u(l, t) = 0 (t > 0) u(x, 0) = ϕ(x), u t (x, 0) = ψ(x) (0 x L). u = u(x, t) x t ( ) T ρ ( ) c c = T/ρ ( ) u(x, t) = n=1 sin nπx L ( a n cos cnπt L + b n sin cnπt ), L a n = 2 L L 0 ϕ(x) sin nπx L dx, b n = 2 cnπ L 0 ψ(x) sin nπx L dx. 80
82 ( ) n 2L nc, nc 2L. c 2L (n = 1 ) ( ) (C, C#, D, D#, E, F, F#, G, G#, A, A#, B) ( ) 2 1/12 = A ( ) 440 Hz C ( ) ( 9 ) 440 = Hz. 29/ = C 129 ( ) ( ) T = 1 s f u(t) = e ift 0 t T ( T ) Fourier c n = 1 T T 0 in u(t)e T t dt = 1 T T 0 e i(f n/t )t dt = 1 T T 0 e iant dt = 1 ia n T ( e ia nt 1 ). A n := (f n/t ). c n = sinc A nt 2. T = 1 s, f = Hz n = 125,, 135 sinc(a n T/2) ( ) 81
83 4.4 Mathematica Mathematica Fourier[ ] (C n ) C n := 1 x j ω nj N j=0 C n FourierParameters->{-1,-1} c=fourier[tb, FourierParameters->{-1,1}]; C n = 1 N C n 5 C n 2 = 1 N C n 2 ( n ) ( ) 6 Import[], ListPlay[], Fourier[] 5 x j ω nj = x j ω nj = x j ω nj
84 5 ( ) Fourier ( 3.2.6) f : R C [18] ( ) ( ) 1 (Harry Nyquist, , ) (Calude E. Shannon, ) [4] 3 (1949 ) [19] (1949 ) Kotel nikov 1933 E. T. Whittaker ( , ) 1915 ( , ) 1920 (Butzer [20] Whittaker Whitakker ) [21] 1 ( ) 2 Certain topics in telegraph transmission theory W 1 2W 3 Communication in the presence of noise 83
85 5.0.2 (, Nyquist, Shannon, ) x: R C Fourier X(ω) = 1 x(t)e iωt dt ( W > 0)( ω R : ω W ) X(ω) = 0 W T := π W ( t R) x(t) = n= sin π(n t/t ) x(nt ) π(n t/t ) = n= x(nt ) sinc [π(n t/t )]. W f s := 1 T = W π f 2f Fourier X(ω) := 1 x(t)e iωt dt (ω R) ω W X(ω) = 0 x(t) = 1 X(ω)e iωt dω (t R). (5.1) x(t) = 1 W W X(ω)e iωt dω (t R) 2W Fourier X(ω) ( ω W ) Fourier (d n = c n ) c n := 1 2W X(ω) = n= W W X(ω)e in π W ω dω c n e in π W ω ( ω W ). d n := 1 2W W W X(ω)e in π W ω dω (5.2) X(ω) = d n e in π W ω ( ω W ) n= 84
86 d n (5.1) d n = 1 T := π W. π W 1 W W X(ω)e iω n π W dω = 1 T x(nt ), (Fourier X(ω) Fourier d n ) (5.2) : X(ω) = (5.1) x(t) = 1 ( W T W n= T n= x(nt )e inωt ) x(nt )e inωt ( ω W ). e iωt dω = T n= W x(nt ) e iω(t nt ) dω. W W W e iω(t nt ) dω = eiw (t nt ) iw (t nt ) e i (t nt ) = 1 2 sin π(n t/t ). T n t/t = 2 sin W (t nt ) t nt = 2 sin π (nt t) T T (n t/t ) ( 1 2a a a e ixξ dx = sin(aξ) aξ x(t) = ) n= sin π(n t/t ) x(nt ) π (n t/t ). ( ) 85
87 6 Fourier Fourier 6.1 Fourier {f n } n Z {f n } n Z f n = f(n) f : Z C C Z f Fourier (discrete-time Fourier transform, DTFT) (6.1) Ff(ω) = f(ω) := f f f(ω + ) = n= f(n)e in(ω+) = n= n= f(n)e inω (ω R) f(n)e inω i2nπ = n= f(n)e inω = f(ω). ω [0, ] ( ω [, π]) f(n) f ( n) Fourier ( ) (6.2) f(n) = 1 f(ω)e inω dω (n Z) Fourier m n ( e inω, e imω) = e inω e imω dω = [ e e i(m n)ω i(m n)ω dω = i(m n) ] π = 0 {e inω } (6.1) f e inω f(n) ( e inω, e inω) = e inω e inω dω = dω = f(n) = (f, e inω ) (e inω, e inω ) = f(ω)e inω dω = 1 f(ω)e inω dω. 86
88 {f(n)} n Z n= f(n) 2 < ( {f(n)} l 2 (Z) ) f L 2 (0, ) (6.1) L 2 {f(n)} n Z n= f(n) < ( {f(n)} l 1 (Z) ) (6.1) (Weierstrass M-test ) 6.2 Fourier Fourier Fourier (Fourier ) Fourier Fourier Fourier Fourier (Fourier ) Fourier (discrete Fourier transform) Fourier (discrete-time Fourier transform) R Fourier f(ξ) = 1 R Fourier c n = 1 Z ( ) Z ( ) Fourier Fourier f(ω) = C n = 1 N 0 n= j=0 ω := exp f(x)e ixξ dx (ξ R) f(x) = 1 f(x)e inx dx (n Z) f(x) = f(n)e inω (ω [0, ]) f(n) = 1 f j ω nj (0 n N 1), f j = ( ) i N n= n=0 0 C n ω nj c n e inx f(ξ)e ixξ dξ f(ω)e inω dω ( ) ( ) L 2 (R) L 2 (R), L 2 (0, ) l 2 (Z), l 2 (Z) L 2 (0, ), C N C N ( 1 1 N ) Fourier Fourier ( ) C N ( L 2 (R) Fourier ) Fourier Fourier Fourier ( 2 3 ) Fourier, Fourier Fourier, Fourier R R Fourier 87
89 7 7.1 (, ) (, convolution) f g (7.1) (7.2) (7.3) (7.4) (7.5) f g = g f, ( ) (f g) h = f (g h), ( ) (f 1 + f 2 ) g = f 1 g + f 2 g, ( ) (cf) g = c(f g) ( ) [f 0 f g = f h] g = h ( ) ( ) Fourier Fourier Fourier (7.6) F[f g] = FfFg. ( Fourier ) δ (7.7) f δ = f. δ ( ) δ δ = {δ n0 } n Z = {, 0, 0, 1, 0, 0, } δ Fourier 1 1 Fourier δ (7.8) Fδ = 1, F1 = δ. Fourier R Fourier Fδ(ξ) = 1, F1(ξ) = δ(ξ) f Ff = F[f δ] = FfFδ F 1(x) = δ(x) Fourier Fourier F1(ξ) = δ( ξ) = δ(ξ) 88
90 7.2 f g f g (, the convolution of f and g) f, g : R C f g : R C (7.9) f g(x) := f(x y)g(y) dy (x R) f, g : R C f g : R C (7.10) f g(x) := 1 f, g : Z C f g : Z C (7.11) f g(n) := k= N f, g : Z C f g : Z C (7.12) f g(n) := k=0 N f(x y)g(y) dy (x R) f(n k)g(k) (n Z) f(n k)g(k) (n Z) 7.3 ( ) Fourier Dirichlet f : R C n s n (x) = a n (a n cos nx + b n sin nx) = k=1 k= n Dirichlet n (7.13) D n (x) := k= n (7.14) s n = D n f. 89 e ikx n c k e ikx
91 s n (x) = n k= n c k = 1 f(x)e ikx dx 1 π f(y)e iky dye ikx = 1 n k= n e ik(x y) f(y)dy = D n f(x) (Dirichlet ) n N, θ R \ {2nπ n Z} D n (θ) = k=1 n k= n D n (θ) = e ikθ = 1 + sin [(n + 1/2)θ]. sin(θ/2) n ( e ikθ + e ikθ) = Re k=1 n e ikθ. e iθ 1 n e ikθ = e iθ einθ 1 e iθ 1 = ei(n+1/2)θ e iθ/2 = ei(n+1/2)θ e iθ/2. e iθ/2 e iθ/2 2i sin(θ/2) k=1 Re z i = Im z D n (θ) = Im ( e i(n+1/2)θ e iθ/2) 2 sin(θ/2) = 1 + sin [(n + 1/2)θ] sin(θ/2) sin(θ/2) = sin [(n + 1/2)θ]. sin(θ/2) Mathematica Dirichlet ( ) Di[n_,x_]:=Sin[(n+1/2)x]/Sin[x/2]; Di2[n_,x_]:=Sum[Exp[I k x],{k,-n,n}]; g=plot[{di[4_,x_],di2[4,x]},{x,-3pi,3pi}] (Dirichlet [, π] ) f lim n s n = f ( ) lim n D n = δ ( ) 1 ( ) E (x) = 1 x 4π x 3. (SI 1 ε 0 E (x) = 1 x 4πε 0 x 3 ) 1 ε 0 = 107 4πc 2. c c = m/s. ε F/m. 90
92 7.1: Dirichlet D n (n = 4 ) 2 1 (, ) U U(x) = 1 1 4π x. grad U = E E u grad u = E ( ) U 2 ( ) Q u(x) = QU(x). y Q u(x) = QU(x y). y 1, y 2,, y N Q 1, Q 2,, Q N u(x) = N Q j U(x y j ). j=1 q(y) (7.15) u(x) = U(x y)q(y) dy. R 3 R 3 f, g : R 3 C f g f g(x) = f(x y)g(y) dy R 3 (7.15) u = U q (U ) 2 91
93 q U q ( U ) Gauss 3 ( ) div E = q E = grad u div grad = u = q. Poisson ( ) Dirac δ (7.16) U = δ. ( 1 ) ((7.16) U ) (f g) = f g = f g. x j x j x j (f g) = ( f) g. (f g)(x) = f(x y)g(y) dy = f(x y)g(y) dy. x j x j R 3 R x 3 j δ q q δ = q U = δ u := U q u u = (U q) = ( U) q = δ q = q. U 1 ( ) F h := F [δ] x F [x] h x 3 Maxwell 1 div E = ρ ε 0 ρ ε 0 = 1 92
94 7.3.3 ( ) ( ) 7.4 f : R C 4 Fourier 4 ( ) f g f (7.17) (7.18) (f 1 + f 2 ) g = (f 1 g) + (f 2 g), (cf) g = c(f g) (c C) f g = g f f : R C, g : R C f g(x) = f(x y)g(y) dy. u = x y du = dy, y = u =, y = u =, y = x u f g(x) = f(u)g(x u)( du) = N f : Z C, g : Z C f g(n) = k=0 f(n k)g(k). g(x u)f(u) du = g f(x). l := n k (k l ) 0 k N 1 n l n (N 1) n f g(n) = g(n l)g(l). l=n () 4?? F
95 g(n l)f(l) l N N f g(n) = l=0 g(n l)g(l) = g f(n) (f g) h = f (g h) (Fubini ) (g f) h(x) = = = = (g f)(x y)h(y)dy ( ) g ((x y) u) f(u) du h(y) dy ( ) g ((x u) y) h(y) dy f(u) du g h(x u)f(u) du = (g h) f(x). (g f) h = (g h) f. (f g) h = (g f) h = (g h) f = f (g h) (the Titchmarsh convolution theorem) (7.19) f g = 0 f = 0 g = 0. 5 ( ) Yosida [22] ( ), [9] ( ) 7.5 Fourier Fourier f g Fourier Ff F[f g] = (Ff Fg) ( Fourier Fourier ) (f, g f g f Ff ) 5 0 f f f δ f 94
96 Z {f j } j Z f j f(j) f : Z j f j C Fourier ( ) ( ) Fourier f R f : R C f Fourier Ff ( f ) Ff(ξ) = f(ξ) := 1 f(x)e ixξ dx (ξ R) Ff : R C Fourier f, g : R C f g f g f g(x) := f g : R C f(x y)g(y)dy (x R) (7.20) F[f g](ξ) = Ff(ξ)Fg(ξ) h := f g F[f g](ξ) = 1 = 1 = 1 = 1 h(x)e ixξ dx ( ) f(x y)g(y)dy e ixξ dx ( ) f(x y)g(y)e ixξ dx dy ( ) f(x y)e ixξ dx g(y)dy. ( ) lim u = x y dx = du, x = u + y, e ixξ = e i(u+y)ξ = e iuξ e iyξ R R y f(x y)e ixξ dx = lim f(x y)e ixξ dx = lim R R R R y = e iyξ lim R R y R y f(u)e iuξ du = e iyξ f(u)e iuξ du. f(u)e iuξ e iyξ du 95
97 ( y ) F[f g](ξ) = 1 ) (e iyξ f(u)e iuξ du g(y)dy = 1 f(u)e iuξ du = Ff(ξ)Fg(ξ). g(y)e iyξ dy ( ) x y = z x z ( ) ( ) f(x y)g(y)e iξx dy dx = f(z)g(y)e iξ(z+y) dy dz y x y = z ( ) y ( ) f(x y)g(y)e inx dy dx = f(z)g(y)e in(z+y) dz y Fourier Fourier f : R C c n := 1 f(x)e inx dx (n Z) f Fourier ( ) f Fourier Ff f ( ) Ff(n) = f(n) := 1 f(x)e inx dx (n Z) Ff : Z C f, g : R C f g f g f g(x) := 1 f(x y)g(y)dy (x R) f g : R C (7.21) F[f g](n) = Ff(n)Fg(n). h := f g F[f g](n) = 1 = 1 = 1 () 2 = 1 () 2 h(x)e inx dx ( 1 π ) f(x y)g(y)dy e inx dx ( ) f(x y)g(y)e inx dx dy ( ) f(x y)e inx dx g(y) dy. 96
98 ( ) u = x y dx = du, x = u = y, x = π u = π y, x = u + y, e inx = e in(u+y)n = e inu e iny f(x y)e inx dx = y y y f(u)e in(u+y) du = e iny f(u)e inu du. u f(u)e inu [ y, π y] [, π] F[f g](n) = 1 ) (e iny () 2 f(u)e inu du g(y) dy = 1 = Ff(n)Fg(n). f(u)e inu du 1 y g(y)e iny dy Fourier Fourier N N ω := e i/n N {f j } j Z C n := 1 N j=0 ω nj f j (n Z) {f j } j Z Fourier Fourier f : Z C N ( N ) Ff(n) = f(n) := 1 N j=0 f(j)ω nj (n Z) Ff ( ) f Fourier Ff : Z C N N ( N ) f, g : Z C f g f g f g(n) := k=0 f g : Z C N f(n k)g(k) (n Z) (7.22) F[f g](n) = NFf(n)Fg(n). h := f g F[f g](n) = 1 N = 1 N = 1 N j=0 j=0 k=0 h(j)ω nj ( k=0 ( j=0 f(j k)g(k) ) ω nj = 1 N f(j k)ω nj ) g(k). k=0 ( j=0 f(j k)g(k)ω nj ) 97
99 ( ) ( ) l = j k j = 0 l = k, j = N 1 l = N 1 k, j = l + k, ω nj = ω in(l+k) = ω nl ω nk j=0 f(j k)ω nj = k l= k k f(l)ω nl ω nk = ω nk l= k f(l)ω nl. l f(l)e nl N l = k, k + 1,, N 1 k l = 0, 1,, N 1 F[f g](n) = 1 N k=0 ( k l= k ω nk k = NFf(n)Fg(n). l= k f(l)ω nl = f(l)ω nl ) l=0 f(l)ω nl. g(k) = 1 N l=0 f(l)ω nl k=0 g(k)ω nk Fourier Fourier {x n } n Z X(ω) := x n e inω (ω R) {x n } n Z Fourier n= Fourier f : Z C Ff(ξ) = f(ξ) := f(n)e inξ (ξ R) n= Ff = f ( ) f Fourier Ff : R C f, g : Z C f g f g : Z C f g(n) := f(n k)g(k) (n Z) k= (7.23) F[f g](ξ) = Ff(ξ)Fg(ξ). h := f g ( ) F[f g](ξ) = h(n)e inξ = f(n k)g(k) n= n= k= ( ) = f(n k)e inξ g(k) = k= n= ( ) ( ) = f(l)e ilξ g(k)e ikξ l= k= = Ff(ξ)Fg(ξ). e inξ ( ) f(l)e ilξ e ikξ g(k) k= l= 98
100 7.5.5 ( ) Fourier Fourier F [f g] = F ff g f : R C Fourier Ff(ξ) = 1 Fourier g : R C F g(x) = 1 F g f(x)e ixξ dx (ξ R) g(ξ)e ixξ dξ (x R) (7.24) F [f g](x) = F f(x)f g(x) (7.20) f : R C Fourier (Fourier ) Ff(n) = 1 f(x)e inx dx Fourier g = {g(n)} n Z F g(x) = F g n= g(n)e inx (x R) (7.25) F [f g] (x) = F f(x)f g(x) Fourier (7.23) F [f g] (ξ) = Ff(ξ)Fg(ξ) N f = {f(n)} n Z Fourier ( Fourier ) Ff(n) = 1 N j=0 f(j)ω nj Fourier g = {g(n)} n Z (7.26) F g(j) := F g n=0 g(n)ω nj (j Z) (7.27) F [f g] = F ff g Fourier (7.22) F [f g] (n) = NFf(n)Fg(n) 99
101 f = {f(n)} n Z Fourier ( Fourier ) Ff(ω) = n= f(n)e inω Fourier g : R C F g(n) = 1 F g g(x)e inx dx F [f g] = F ff g Fourier (7.21) F [f g] (n) = F f(n)f g(n) 7.6 ( ) d df (f g) = dx dx g = f dg dx. f g(x) = d (f g(x)) = dx f(x y)g(y) dy = x f(x y)g(y) dy f (x y)g(y) dy = (f ) g(x). 100
102 8 Z N N = {n Z n 1} = {1, 2, 3, }. A B B A ( ) 8.1 a = {a n } n N N C (1 ) ( a n a C N n N a(n) ) C N ( ) {a n } n Z C Z S := C Z ( signal ) (8.1) S = C Z =. (S S calligraphy ) x, y S, c C (x + y)(n) := x(n) + y(n), (cx)(n) := cx(n) (n Z) x + y, cx S S C ( ) S (a discrete signal, a discrete-time signal) 8.2 x, y S x y x y S x y(n) = x(n k)y(k) (n Z) ( ) k= 101
103 1. x y = y x, (x y) z = x (y z) δ = {δ n } n Z S (8.2) δ n := δ n0 = { 1 (n = 0) 0 (n Z \ {0}) δ (the unit impulse) x S (8.3) x δ = δ x = x n Z δ x(n) = δ(n k)x(k) = δ x = x. k= k= (δ n0 Kronecker ) δ n k,0 x(k) = k= δ nk x(k) = x(n) 8.3 (LTI ) S S F : S S x S F [x] 1 F : S S ( x, y S) F [x + y] = F [x] + F [y], ( x S)(c C) F [cx] = cf [x] x = {x n } n Z S, k Z y(n) := x n k = x(n k) (n Z) y S x( k) {x n k } n Z k ( ) F : S S (, time-invariant) x S, k Z (8.4) F [x( k)] = F [x]( k) {y n } n Z := F [{x n } n Z ] ( k Z) F [{x n k }] = {y n k } ( S k [x] = x( k) F S k = S k F ) F : S S h := F [δ] x S F [x] = h x. F h := F [δ] F (the unit impulse response) 1 x x(n) 102
104 x S x(n) = x(k)δ(n k) (n Z) k= k= x = F [ ] F [x] = F x(k)δ( k) = = k= x(k)h( k). F [x](n) = F [x] = h x. k= k= k= x(k)δ( k). x(k)f [δ( k)] = k= x(k)h(n k) = h x(n) (n Z) x(k)f [δ]( k) ( ) ( ) f f 8.4 FIR F FIR (, a finite impulse response filter) F h := F [δ] J ( ) ( n Z : n < 0 n > J) h n = 0 h 0, h 1,, h J ( 0 ) F F [x] = x h F [x](n) = x(n k)h k = k= J x(n k)h k (n Z) k=0 8.5 Mathematica piano-cutoff.nb 2 ( Mathematica 9 Mathematica )
105 piano-do-mi-so.wav Fourier 0 Fourier piano-cutoff.nb snd = Import[" {left, right} = snd[[1, 1]]; sr = snd[[1, 2]] take[tbl_, t1_, t2_] := Take[tbl, {Floor[t1*sr], Floor[t2*sr]}] take1[tbl_, t_] := Take[tbl, {Floor[t*sr], Floor[t*sr] + sr - 1}] g = ListPlot[tbl = take1[left, 1.0], PlotRange -> All] ListPlay[tbl] ListPlay[tbl, SampleRate -> sr] ListPlay[tbl, SampleRate -> sr/2] ListPlay[tbl, SampleRate -> sr*2] ListPlay[tbl, SampleRate -> Floor[sr*1.5]] c = Fourier[tbl, FourierParameters -> {-1, -1}]; g = ListPlot[Abs[c], Joined -> True, PlotRange -> All] cutoff[f_] := Join[Table[1, {n, f + 1}], Table[0, {n, sr - 2*f - 1}], Table[1, {n, f}]]; 440.0*2^(-{9, 5, 2}/12) c2 = c*cutoff[500]; g3 = ListPlot[Abs[c2], Joined -> True, PlotRange -> All] Export["do-mi-so-cutoff500.eps", g3] tbl2 = Re[InverseFourier[c2, FourierParameters -> {-1, -1}]]; ListPlay[tbl2, SampleRate -> sr] do = Re[InverseFourier[c*cutoff[300], FourierParameters -> {-1, -1}]]; domi = Re[InverseFourier[c*cutoff[360], FourierParameters -> {-1, -1}]]; domiso = Re[InverseFourier[c*cutoff[450], FourierParameters -> {-1, -1}]]; domiso2 = Re[InverseFourier[c*cutoff[900], FourierParameters -> {-1, -1}]]; original = Re[InverseFourier[c, FourierParameters -> {-1, -1}]]; ListPlay[do, SampleRate -> sr] ListPlay[domi, SampleRate -> sr] ListPlay[domiso, SampleRate -> sr] ListPlay[domiso2, SampleRate -> sr] ListPlay[original, SampleRate -> sr] Safari WWW lecture/fourier-2017/ piano-cutoff.nb 3 Ctrl+ Mathematica
106 8.5.1 Shift + Enter [ ] [ ] snd = Import[" piano-do-mi-so.wav snd {left, right} = snd[[1, 1]]; sr = snd[[1, 2]] PCM left, right ( ) sr ( Hz ) take[tbl_, t1_, t2_] := Take[tbl, {Floor[t1*sr], Floor[t2*sr]}] take1[tbl_, t_] := Take[tbl, {Floor[t*sr], Floor[t*sr] + sr - 1}] t1 t2 take[] ( ) t 1 take1[] (Take[] Floor[] ) ListPlot[tbl = take1[left, 1.0], PlotRange -> All] tbl : ListPlay[tbl, SampleRate -> sr] 105
107 ListPlay[] ( ) (snd tbl ) SampleRate -> sr /2 ListPlay[tbl, SampleRate -> sr/2] ListPlay[tbl, SampleRate -> Floor[1.5*sr]] ( 1.5 Mathematica Floor[] ) Fourier ( ) Fourier c = Fourier[tbl, FourierParameters -> {-1, -1}]; ListPlot[Abs[c], Joined -> True, PlotRange -> All] Fourier Fourier C n c c[[i]] C i 1 (1 i N = sr = 44100) 1 Fourier (Fonurier ) c n, c n n Hz Fourier C n, C N n c[[n+1]], c[[n-n+1]], 3 3 ( ) 106
108 Hz /12, /12, /12 Hz 440.0*2^(-{9, 5, 2}/12) { , , }, 261.6, 329.6, Hz cutoff[f_] := Join[Table[1, {n, f + 1}], Table[0, {n, sr - 2*f - 1}], Table[1, {n, f}]]; c2=c*cutoff[500]; ListPlot[Abs[c2], Joined -> True, PlotRange -> All] cutoff[] f Hz ( f + 1 1, f 1, 0 ) c2 501 Hz (2 ) Fourier ( ) tbl2 = Re[InverseFourier[c2, FourierParameters -> {-1, -1}]]; ListPlay[tbl2, SampleRate -> sr] 501 Hz ( Mathematica Re[] Mathematica tbl2 0 ListPlay[tbl2, ] ) 107
109 do = Re[InverseFourier[c*cutoff[300], FourierParameters -> {-1, -1}]]; domi = Re[InverseFourier[c*cutoff[360], FourierParameters -> {-1, -1}]]; domiso = Re[InverseFourier[c*cutoff[450], FourierParameters -> {-1, -1}]]; domiso2 = Re[InverseFourier[c*cutoff[900], FourierParameters -> {-1, -1}]]; original = Re[InverseFourier[c, FourierParameters -> {-1, -1}]]; ListPlay[do, SampleRate -> sr] ListPlay[domi, SampleRate -> sr] ListPlay[domiso, SampleRate -> sr] ListPlay[domiso2, SampleRate -> sr] ListPlay[original, SampleRate -> sr] do : 1 ( ) ( ) ( AD ) (LTI ) F DA x(t) x = {x n } n Z y = {y n } n Z. T s ( f s := 1 T s, Ω s = f s ) (8.5) x n = x(nt s ) (n Z). LTI F h := F [δ] (8.6) y = F [x] = x h. 108
110 (8.7) y n = x n k h k k= (n Z) ( x(t) x(t) = 1 x(ξ)e itξ dξ x(t) e itξ (ξ R) ) x(t) = e iωt (Ω R) x n = x(nt s ) = e iωnts = e inω, ω := ΩT s. x n = (e iω ) n x := {x n } n Z e iω T s Ω < Ω s 2 ω < π Ω ω = ΩT s ω (, π) ω (, π], ω ΩT s (mod ) ω [0, ), ω ΩT s (mod ) (ω ΩT 2 ) ω x n = e inω ω ω = f f F, F s f f = ω, ω = ΩT s, Ω = F, T s = 1 F s f = F F s. F = F s f, Ω = F s ω. ( ) = ( ). 109
111 2. F s = 44100Hz ω = 0.1 F Ω = F s ω, Ω = F F = Ω = F sω = 44100Hz Hz F h := F [δ] x F [x] = x h. y n = x n k h k = e i(n k)ω h k = e inω k= k= k= e ikω h k. (8.8) ĥ(ω) := e ikω h k k= (ω [, π]) h (8.9) y n = e inω ĥ(ω) (n Z). y = {y n } ( x ) ĥ(ω) ω F (frequency response), (frequency characteristic) h z h = {h n } z (8.10) H(z) := H(z) k= (8.11) ĥ(ω) = H ( e iω). H(z) F (transfer function) G(ω) := ĥ(ω) = H (e iω ) (gain) θ(ω) := arg ĥ(ω) = arg H (eiω ) (phase shift) ( arg ) h k z k 110
112 8.6.5 ( 8.5 ) F e > 0 F e F e F e ω e (8.12) ĥ(ω) = ω e := F e F s. { 1 ( ω ω e ) 0 ( ω > ω e ) Fourier h n = 1 (8.13) h n = 1 ωe ĥ(ω)e inω dω (n Z) ω e e inω dω = ω e π sinc nω e. sinc sinc (Mathematica Sinc[] Sinc ) sin x (x R \ {0}) sinc x := x 1 (x = 0). 3. (8.13) : h n n= h n e inω = ĥ(ω) = { y n = k= x n k h k 1 ( ω ω e ) 0 ( ω > ω e ) y n F e F e J N ĥ J (ω) := J/2 n= J/2 h n e inω ĥj(ω) 111
113 naivelowpass.nb ĥj(ω) omega=0.5 h[n_]:=omega/pi Sinc[n omega] draw[j_]:=plot[sum[h[n]exp[-i n t],{n,-j/2,j/2}],{t,-pi,pi}, PlotRange->All] draw[100] Plot[If[Abs[x]<omega,1,0],{x,-Pi,Pi}] Gibbs : ĥj(ω) (J = 100), (8.12) {h n } { h h n ( n J/2) n := 0 ( n < J/2). {h n} windowing ( ) h n ( ) ( 0 ) 0 hann w(x) := 1 cos x 2 (0 x 1) hann w[x_]:=(1-cos[2pi x])/2 g=plot[w[x],{x,0,1}] w h n := { w(n/j 1/2)h n ( n J/2) 0 ( n > J/2) {h n} {h n } {h n} 112
114 : w w[x_]:=(1-cos[2 Pi x])/2 draw2[j_]:=plot[sum[w[n/j-1/2]h[n]exp[-i n t],{n,-j/2,j/2}],{t,-pi,pi}, PlotRange->All] : y n = k= x n k h k = J/2 k= J/2 x n k h k. y n x n J/2, x n J/2+1,..., x n+j/2 {h n } {h n} FIR ( ) 113
115 9 2 ( ) 9.1 : Fourier ( 1 Fourier CT 2 Fourier Fourier ) f, g : R n C f Fourier Ff, g Fourier ( Fourier ) F g Ff(ξ) = f(ξ) 1 = f(x)e ix ξ dx, () n/2 R n F 1 g(x) = g(x) = g(ξ)e ix ξ dx () n/2 R n x ξ x ξ : x ξ = x 1 ξ 1 + x 2 ξ x n ξ n. 1 e ix ξ = e ix 1ξ 1 e ix 2ξ2 e ixnξn Ff f = f(x 1, x 2,..., x n ) x j (j = 1, 2,..., n) Fourier 9.2 : ( ) 1 ( ) ( ) : f u (9.1) (9.2) u t (x, t) = u xx (x, t) (x R, t > 0), u(x, 0) = f(x) (x R) ( ) u(x, t) t, x f u(x, t) x Fourier û(ξ, t) : û(ξ, t) = F [u(x, t)] (ξ) = 1 u(x, t)e ixξ dx (ξ R, t > 0). 114
116 u t = u xx Fourier F [u xx (x, t)] (ξ) = (iξ) 2 F [u(x, t)] (ξ) = ξ 2 û(ξ, t), F [u t (x, t)] (ξ) = 1 t u(x, t)e ixξ dx = 1 u(x, t)e ixξ dx = t tû(ξ, t) = ξ2 û(ξ, t). tû(ξ, t) û(ξ, t) = e tξ2 û(ξ, 0) = e tξ2 ˆf(ξ). ( dy dx = ay, y(0) = y 0 y = y 0 e ax ) [ ] ( ) u(x, t) = F e tξ2 ˆf(ξ) (x). 2 ( ) (f Fourier e tξ2 Fourier u ) 1 f, g f g f g(x) := f(x y)g(y) dy (x R) F [f g] = FfFg ( 7 ) f : R C, g : R C, h: R C Fh = gff G := 1F g h = G f. ( ) G = 1 F g Fourier ( ) FG = 1 g. h = FG. Fh = gff Fh = FGFf = F[G f]. Fourier ( ) h = G f. ( ) f g = [ ] F [FfFg] = F Fg ˆf. ( ) F[G(x, t)](ξ) = e tξ 2 G(, t) u(x, t) = G(, t) f(x) ] F [e ax2 (ξ) = 1 [ ] e ξ2 4a F e aξ2 (x) = 1 e x2 4a 2a 2a (9.3) u(x, t) = G(x, t) = 1 F [ e tξ2] (x) = 1 4πt e x2 4t. G(x y, t)f(y) dy (x R, t > 0), G(x, t) = 1 4πt e x2 4t. 115
117 (9.1), (9.2) u ( ) G (fundamental solution), Green (Green function), (heat kernel) G t > 0 G(x, t) > 0, G(x, t) dx = 1 G 0, 2t 1 t x G(x, t) Mathematica G[x_, t_] := Exp[-x^2/(4 t)]/(2*sqrt[pi*t]) g=plot[table[g[x, t], {t, 0.1, 1.0, 0.1}], {x, -5, 5}, PlotRange -> All] Manipulate[Plot[G[x, t], {x, -5, 5}, PlotRange -> {0, 3}], {t, 0.01, 2}] : G(, t) (t = 0.1, 0.2,..., 1.0) G : 2 G(x, t) = G(x, t). t x2 t + G(x, t) 0 t lim G(x, t) = t +0 { 0 (x 0) + (x = 0) t +0 G(x, t) Dirac ( ) : lim G(x, t) = δ(x). t +0 1 m, σ 2 N(m; σ 2 ) 1 ) ( σ 2 exp (x m)2 2σ 2 116
118 lim u(x, t) = f(x) ( ) t +0 ( (9.2) ) ( ) G Dirac : 2 G(x, t) = G(x, t), t x2 G(x, 0) = δ(x). 0 ( ) ( ) 7. ψ : R R (9.4) (9.5) 2 u t (x, t) = 2 u (x, t) 2 x2 u(x, 0) = 0, u ( ) (1) u x Fourier û(ξ, t) = 1 ((x, t) R (0, )), u (x, 0) = ψ(x) (x R) t u(x, t)e ixξ dx (2) û Fourier u ( ( G.1 ) u(x, t) = 1 2 ) x+t x t ψ(y) dy 9.3 ( ) ( ) u 0 S(R) (9.6) (9.7) (9.8) u t (x, t) = u xx (x, t) (x R, t > 0), u(x, t) 0 ( x, t > 0), u(x, 0) = u 0 (x) (x R) u = u(x, t) u = u(x, t) t x Fourier û = û(, t) (9.9) û(ξ, t) := 1 u(x, t)e ix ξ dx (ξ R). 1 u t (x, t)e ix ξ dx = 1 R t R 117 R u(x, t)e ix ξ dx = û (ξ, t). ξ
119 F[f ](ξ) = iξf(ξ) F[f ](ξ) = (iξ) 2 F(ξ) = ξ 2 F(ξ) 1 u xx (x, t)e ix ξ dx = ξ 2 û(ξ, t). (9.6) x Fourier (9.10) R û t (ξ, t) = ξ2 û(ξ, t). (9.8) Fourier ( (9.9) t = 0 (9.8) ) (9.11) û(ξ, 0) = û 0 (ξ). (9.10), (9.11) : Fourier u û(ξ, t) = e ξ2tû 0 (ξ). u(x, t) = 1 f(ξ)e ξ2t e ixξ dξ. ( ) u 0 û 0 F (f g) = FfFg (9.12) û(ξ, t) = 1 e ξ2tû 0 (ξ) = Ĥ(ξ, t)û 0(ξ) = F [H(, t) f] (ξ). [ ] 1 (9.13) H(x, t) := F e ξ2 t (x). Gaussian Fourier [ F e ax2] (ξ) = 1 e ξ2 4a 2a H(x, t) = 1 1 e x2 4t. 2t (9.14) H(x, t) = 1 ) exp ( x2. 4πt 4t H(x, t) (the fundamental solution of the heat equation) (heat kernel) (9.12) û(ξ, t) = F [H(, t) u 0 ] (ξ) Fourier u(x, t) = H(, t) u 0 (x) = : (9.15) u(x, t) = H(x y, t)u 0 (y) dy = 1 4πt 118 H(x y, t)u 0 (y) dy. exp [ ] (x y)2 u 0 (y) dy. 4t
120 (9.15) u H H t (x, t) = H xx (x, t) (x R, t > 0), H(x, 0) = δ(x). H(x, t) t = 0 t = 0 x = 0 t(> 0) x H(x, t) x = 0 x 1 ( ) H(x, 1/2) H(x, t) 0, 2t anim.gp plot [-10:10] [0:1] H(x,t) t=t+dt if (t<tmax) reread anim.gp gnuplot gnuplot> H(x,t)=exp(-x*x/(4*t))/sqrt(4*pi*t) gnuplot> t=0.1 gnuplot> Tmax=5 gnuplot> dt=0.01 gnuplot> load "anim.gp" t = t = 5 H(, t) GIF 9.4 CT ( ) CT (computed tomography, 2 ) G. N. Hounsfield A. M. Cormack 1972 (1979 ) X X ( ) CT Johann Radon 1917 ([23]) 2 Radon 2 tomography X X 1 119
121 H(x,1) H(x,2) H(x,3) H(x,4) : H(, t) t = 1, 2, 3, 4 9.3: X X 120
122 f Rf f Radon Radon Rf sinogram 3 f : R 2 R ( 0 ) R 2 L Rf(L) := f(x) dx (L ) L 9.4: L s( 0) x θ ( [0, ]) L ( ) ( ) ( ) x(t) s cos θ sin θ = + t (t R) y(t) s sin θ cos θ : Rf(θ, s) = f(x(t), y(t))dt = f(s cos θ t sin θ, s sin θ + t cos θ)dt. ( ) Rf(θ, s) f(x, y) f : R 2 R Rf(θ, X) := Rf f Radon f(x cos θ Y sin θ, X sin θ + Y cos θ)dy (θ [0, ], X R) 3 (Dirac Radon 121
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I kawazoe@sfc.keio.ac.jp chap. Fourier Jean-Baptiste-Joseph Fourier (768.3.-83.5.6) Auxerre Ecole Polytrchnique Napoleon G.Monge Isere Napoleon Academie Francaise [] [ ] [] [] [ ] [ ] [] chap. + + Fourier
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