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2 ( ) 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:- 1

3 N = {1, 2, } N 0 = {0, 1, 2, } Z = {0, ±1, ±2, } Q = R = C = P (p. 17) δ mn (p. 21) z z Re z, Im z z Span φ 1,..., φ N (p. 119) f Fourier F f, Ff, f, F[f(x)](ξ) F, F F ( F ) sinc x := sin x x ( Wikipedia [sínk]) 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 {a n } n Z a: Z C C Z 2

4 Fourier ( + ) ( = ) Fourier Bessel, Parseval, Fourier Fourier Fourier, Fourier Fourier Fourier Fourier e ax2 Fourier : Fourier x 2 + a Mathematica Fourier Fourier Fourier Fourier Fourier : ( ) : ( )

5 3 Fourier Fourier Fourier 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

6 7.5.2 Fourier Fourier Fourier Fourier Fourier Fourier (LTI ) FIR Fourier : : Fourier CT A 117 B misc 119 B.1 : (Bessel, Schwarz, ) C Fourier 123 C.1 Fourier C D Hilbert 127 D D E Fourier 130 E.1 Lebesgue E E.1.2, E

7 E.1.4 Lebesgue E.2 Lebesgue E E E.3 Lebesgue Fourier E.4 Lebesgue Fourier F ( ) 136 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 142 G.1 1 R G.1.1 d Alembert G.1.2, d Alembert H [0, ) Laplace 146 I Fourier 147 I J 153 K Laplace z 154 K.0.1 Laplace K.1 z L sinc 156 L.1 :

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 : ( ) ( ) ( ) (,,,, ) ( ) ( ) ( ) ( L 2 ) ( ) ( ) Lebesgue 9

11 1 Fourier ( + ) Fourier (?) ( ) 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 10

12 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 nx α+ cos nx ( α R) f(x) sin nx e dx = f(x) sin nx inx α e dx. inx [, π] [0, ] ( ) [, π] 11

13 (, 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 a a ( )dx = 0, a ( )dx = 2 a a 0 ( )dx Fourier x f ( ) ( 1.4) ( N n= N c n e inx ) s N (x) := a N (a n cos nx + b n sin nx) n=1 12

14 (1) f C 1 f Fourier f ( s N ( ) f ): lim sup f(x) s N (x) = 0. N ( 1.4.3) x R (2) f ( C 1 ) ( ) f Fourier ( ) f f lim s N(x) = N f(x) f(x 0) + f(x + 0) 2 (x f ) (x f ). Gibbs ( ) 1 (3) f ( f 2 ) f : lim N f(x) s N (x) 2 dx = 0. (?) Lebesgue 1.2, 1.3 (4) f Fourier 2 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=

15 ( ) a n (z c) n ( n=0 ) (1.3) (Taylor ) Fourier ( 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 n 1 sin nx ( 1), g(x) = 4 ( 1) n 2 n n=1 n=1 (x R). 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}] f f n : f 1.2: s 10 s n f g g Fourier s n 14

16 1.3: n f 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->100,PlotRange->{-8,8}], {n,1,20}] ( g Fourier, PlotPoints->100 ) g D := {(2k 1)π k Z} g g R \ D 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 15

17 : g 1.5: s : n g s n 16

18 Gibbs 1.2 Fourier ( ) Fourier Φ := {cos nx} n 0 {sin nx} n N Φ := {e inx } n Z ( ) 0 ( ) Φ (1.9) sin mx sin nx dx = 0 cos mx cos nx dx = 0 cos mx sin nx dx = 0 (m, n N, m n), (m, n N {0}, m n), (m N {0}, n N). (1.10) e inx e imx dx = 0 (n, m Z, n m). ( 1 + 2i = 1 2i. e imx = cos(nx) + i sin(nx) = cos(nx) i sin(nx) = e imx ) m, n Z, m n cos mx cos nx dx = 0 (1.11) P := {f f : R C } ( (periodic) P, = ) C ( ) f, g P f g (f, g) (1.12) (f, g) := f (norm) f f(x)g(x) dx (1.13) f := (f, f) 3 17

19 (i) f X f 0. f = 0 (ii) f X, λ C λf = λ f. (iii) f, g X f + g f + g. (f, g) f g f, g (f g) ( ) X = P (1.12) (, ) (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, λ 1, λ 2 C (λ 1 f 1 + λ 2 f 2, g) = λ 1 (f 1, g) + λ 2 (f 2, g). (i) (ii) (f, f) = f(x)f(x) dx = f(x) 2 dx 0. (f, f) = 0 f(x) 2 = 0 f(x) = 0. ( x f(x) = 0 0 Lebesgue ) (f, g) = (iii) ( ) f(x)g(x) dx = f(x)g(x) dx = 4. (ii), (iii) (a) f, g 1, g 2 X µ 1, µ 2 C f(x) g(x) dx = (f, µ 1 g 1 + µ 2 g 2 ) = µ 1 (f, g 1 ) + µ 2 (f, g 2 ). g(x)f(x) dx = (g, f). (b) f, g X f + g 2 = f Re (f, g) + g 2. (Re Re(1 + 2i) = 1.) (, ) C X (i), (ii), (iii) (, ) (, ) X X C (i), (ii), (iii) 18

20 ( ) P C N C N C C N ( ) (x, y) = N x j y j j=1 P (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 X (i), (ii), (iii) (, ) (, ) R X C (i), (ii), (iii) R N R ( ) C N ( ) ( ) (x, y) = N x j y j j=1 (i) (x, x) 0 ( ) ( ) ( ) ( ) X P C R ( X = C n X = R n ) 19

21 1.2.4 ( ) X a, b X a b ( (a, b) = 0) a + b 2 = a 2 + b 2. ( 2 = = ) 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. 20

22 (f, g) < f g (R ( ) C λ = te iθ ) 6. ( ) (i) (i ) f X (f, f) 0. ( (f, f) = 0 f = 0 ) Schwarz Gram-Schmidt (, ) 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) ) 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 ( ) {φ n } n X ψ n := 1 φ n φ n {ψ n } n X ( {ψ n } {φ n } ) 21

23 ( 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 ) {cos nx} n 0 {sin nx} n N = {1, cos x, sin x, cos 2x, sin 2x,, cos kx, sin kx, } X = P k N (cos kx, cos kx) = (sin kx, sin kx) = cos 2 kx dx = sin 2 kx dx = cos kx = π, sin kx = π. k = 0 cos kx = cos 0 = 1 (cos kx, cos kx) = 1 + cos 2kx dx = π, 2 1 cos 2kx dx = π. 2 dx =, cos kx = 1 =. { 1, cos x, sin x } cos 2x sin 2x cos kx sin kx,,,,, π π π π π π P { e inx } n Z = { 1, e ix, e ix, e 2ix, e 2ix,, e ikx, e ikx, } P { } e inx n Z = { 1 1, P 1 e ix, 1 e ix, 1 e 2ix, 1 e 2ix,, 1 e ikx, } 1 e ikx, ( ) X f = n c n φ n (1) {φ n } (2) {φ n } c n = (f, φ n) (φ n, φ n ). c n = (f, φ n ). 22

24 (1) n ( ) (f, φ n ) = c m φ m, φ n = m m c m (φ m, φ n ) = c n (φ n, φ n ). (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 ) n ( ) ( ) lim (f n, g n ) j = lim f n, lim g n n n n ( ) X = P, φ n (x) = e inx (n Z) (φ m, φ n ) = δ mn f = c n φ n n= n N ( ) a n = b n n = 0 c n = (f, φ n) (φ n, φ n ) = 1 f(x)e inx dx = 1 f(x)e inx dx. (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 cos nx = cos 0x = 1 (cos nx, cos nx) = (1, 1) = dx = (f, cos nx) (cos nx, cos nx) = f(x) dx = f(x) dx = a 0 π 2. 23

25 1.3 ( = ) ( ) 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 = K = R K = C. n=1 c n φ n c 1,..., c N K l = V F, l G, F l F H. π = V F, π G, F π F H. G F F H (H) (V ) 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 24

26 (1) (f h) V h V f h = inf f g ( ). g V (2) f h = inf f g ( ) h V (f h) V. g V ( : D.1 ) (1) (1) g V f g 2 = f h 2 + g h 2 f g f h. f h (2) I[g] := f g 2 (g V ) I g = h V g v := g h t K h + tv V f(t) := I[h + tv] (t K) f 2 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. f(0) = I[h] I 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. g V (f h, g h) = (f h, v) = 0 (f h) V 25

27 ( : (Hilbert ) (, D.1 ) ) φ 1,..., φ N X V = Span φ 1,, φ N f X f h = inf f g g V h V h V h = N c n φ n c 1,..., c N ( ) c n (f h) V c n = (h, φ n) (φ n, φ n ) j=1 (n = 1,, N) (f h, φ n ) = 0 (n = 1,, N) (f, φ n ) = (h, φ n ) (n = 1,, N). (1.16) h = c n = (f, φ n) (φ h, φ n ) N n=1 φ 1,, φ N (n = 1,, N). (f, φ n ) (φ n, φ n ) φ n ( ). N (1.17) h = (f, φ n ) φ n n=1 ( ) Fourier Fourier ( ) {φ n } f f = n c n φ n ( f Fourier ) c n c n = (f, φ n) (φ n, φ n ) 26

28 Fourier ( ) s N := N c n φ n n=1 N f f s N : f Fourier s N f V N := Span φ 1, φ 2,..., φ N ( ) f s N = inf g V N f g ( ) s N ( ) s N V N s N f f s N (, ) s N f s N V N f s N s N = N n=1 c n φ n, c n = (f, φ n) (φ n, φ n ) Bessel, Parseval, Fourier Bessel Fourier ( ) Bessel {φ n } n N X f X (1.18) c n 2 f 2, c n := (f, φ n ) n=1 (f, φ n ) 2 f 2 n=1 s N 2 f 2 ( N N s N f ) 0, s N, f 3 f 2 = s N 2 + f s N 2 27

29 ( ) N s N = c n φ n n=1 N 2 c n φ n f 2. n=1 {φ n } ( ) 3 N c n 2 φ n 2 f 2. n=1 ( φ n = 1) N c n 2 f 2. N N n=1 c n 2 f 2. n=1 Bessel (1.18) 0 lim c n = 0. n Riemann-Lebesgue 4 N c n 2 = s N 2 = f 2 f s N 2 n=1 ( ) lim f s N = 0 N ( ) c n 2 = f 2. n=1 f ( ) {φ n } {φ n } ( ) 3 N 2 N N N N j k (φ j, φ k ) = 0 c k φ k = c k φ k, c j φ j = c k c j (φ k, φ j ) = N N c k c k (φ k, φ k ) = c k 2 φ k 2. k=1 k=1 k=1 k=1 j=1 k=1 j=1 4 f lim n (f, φ n) = 0 Riemann-Lebesgue f 2 ( f ) 28

30 ( ) Parseval ( Bessel Parseval ) X (1) {φ n } n N X f X (f, φ n ) 2 f 2 n=1 (Bessel ). lim (f, φ n) = 0 n ( Riemann-Lebesgue ). (2) {φ n } n N X f X (f, φ n ) 2 = f 2 n=1 (Parseval ). Bessel Parseval (( ) Bessel, Parseval ) X {φ n } n N X f X (f, φ n ) 2 φ n 2 f 2. n=1 {φ n } n N 1 ψ n := φ n 2 φ n {φ n } X Bessel (f, ψ i ) 2 f 2. φ n n= (Schwarz Bessel ) g 0 φ 1 := g {φ 1 } Bessel 1 (f, φ k ) 2 φ 1 2 f 2 k=1 (f, g) 2 g 2 f 2. (f, g) 2 f 2 g 2. Schwarz 29

31 (1.19) f = c n φ n n=1 ( ) f s N 0 N (1.20) lim N f c n φ n = 0 {φ n } n N X f X n=1 f = P (f, g) = (f, φ n )φ n. n=1 { } { } 1 1 π cos nx n N f(x)g(x)dx { } 1 π sin nx n N { 1 e inx } n Z 1.4 f : R C Fourier (1.21) f(x) = a (a n cos nx + b n sin nx) = c n e inx (x R). n=1 n= 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 N 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), (1.21) ( ) f (x) (1.22) f (x) =?? ( na n sin nx + nb n cos nx) = inc n e inx (x R). n=1 n= (1.21) (1.22) 30

32 f Fourier f Fourier ( Fourier ) f : R C 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) = n=1 n= inc n e inx 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). (x R). ) f(x)( n sin nx)dx ) f(x)(n cos nx)dx f(x) ( ine inx) ) dx f C 1 {x k } N k=0 f [xk 1,x k ] C 1 c n (f ) = 1 = 1 = 1 N k=1 = in 1 = 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 ( [f(x)e inx ] π + in f(x)e inx dx = inc n (f). xk x k 1 f (x)e inx dx f(x) ( ine inx) dx x k 1 ) f(x)e inx dx ) f ( ) 31

33 c n (f) Fourier F[f](n) (1.23) F[f ](n) = inf[f](n). Fourier F [f ] (ξ) = iξf[f](ξ) f k (1.24) F[f (k) ](n) = (in) k F[f](n). 1 = in 1 Fourier ( ) ( ) f n Fourier Fourier Fourier n ( ) ( Fourier ) (Fourier Parseval ) f : R C (1) f (a) a n, b n 1 π f(x) dx, c n 1 f(x) dx. f(x) M a n, b n 2M, c n M (n Z). (b) lim a n = lim b n = 0, lim c n = 0 (Riemann-Lebesgue ). n n n ± (2) f 2 π ( a n= + ( an 2 + b n 2)) = f(x) 2 dx, n=1 c n 2 = f(x) 2 dx. (1) (a) 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) M f(x) dx f(x) cos nx dx 1 π f(x) dx. f(x)e inx dx 1 f(x) dx. M dx = M. (b) ( ) f 2 (2) 0 32

34 (2) cos nx 2 dx = { π (n N) (n = 0), sin nx 2 dx = π f f f ( C 1 Fourier ) f : R C C 1 f Fourier f f ( C.1.1, p. 123) Fourier inc n f Fourier inc n 2 = f (x) 2 dx. c n e inx = c n c n = n 0 n Z n 0 = Weierstrass M test n= n= 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 1.5 ( windowing ) 33

35 1.6 Fourier ( ) f : [0, π] C C 1 f(x) = a a n cos nx, n=1 a n = 2 π 0 f(x) cos nx dx (Fourier ) f { f(x) (x [0, π]) f(x) := f( x) (x [, 0)) f(x) = a (a n cos nx + b n sin nx), a n = 1 π n=1 f(x) cos nx dx, b n = 1 π f(x) sin nx dx. f [0, π] f a n = 2 π 0 f(x) cos nx dx = 2 π f(x) cos nx dx, b n = 0. f(0) = f(π) = 0 C 1 f : [0, π] C f(x) = b n sin nx, n=1 b n = 2 π 0 f(x) sin nx dx (Fourier ) f Fourier Fourier ( ) ( Dirichlet ) u t (x, t) = u xx (x, t) u(0, t) = u(π, t) = 0 u(x, 0) = f(x) Fourier 5 ((x, t) (0, π) (0, )), (x [0, π]) (t (0, )), u(x, t) = b n e n2t sin nx, n=1 b n = 2 π 0 f(x) sin nx dx. f Fourier u u(x, 0) = f(x) (x [0, π]) 5 u t = u xx u(0, t) = u(1, t) = 0 u(x, t) = ζ(x)η(t) ζ = λζ, ζ(0) = ζ(1) = 0, η = λη 2 λ = n 2 (n N), ζ(x) = C sin nx (C ) η η(t) = C e n2t (C ) u(x, t) = C e n2t sin nx. 34

36 ( Neumann ) u t (x, t) = u xx (x, t) ((x, t) (0, π) (0, )), u x (0, t) = u x (π, t) = 0 (t (0, )), u(x, 0) = f(x) (x [0, π]) u(x, t) = a a n e n2t cos nx, n=1 f Fourier a n = 2 π 0 f(x) cos nx dx. 35

37 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 36

38 f : R C f f(x) x ± (0 ) : l > 0 f [ l, l] Fourier l c n := 1 2l 2 f(x) = n= l l c n e in π l x f(x)e in π l x dx (x [ l, l]). nπ l ( l ) Riemann, b a b a F (x) dx = lim 0 F (x) dx = lim n n F (ξ j ) x j, ξ j [x j 1, x j ], x j := x j x j 1, j=1 n F j=1 ( j b a ) b a n n ( n ) (2.1) F (ξ) dξ = lim ξ 0 n= F (n ξ) ξ ( x ξ j n ) π ( ξ) (2.1) l c n := l π c n = 1 l f(x)e in π l x dx l f(x) = n= c ne in π l x π l. f f Fourier (the Fourier transform of f) f = Ff (2.2) f(ξ) = Ff(ξ) := 1 f(x)e iξx dx (ξ R) ( [ l, l] (, ) ) F : f f Fourier (Fourier transform, Fourier transformation 3 ) 2 : φ n (x) := e i nπ l x {φ n } [ l, l] f = n c n φ n c n = (f, φ n) l π (φ n, φ n ) 3 transform transformation Ff the Fourier transform of f transform ( ) F Fourier tansformation 37

39 f [ l, l] 4 c n = 1 l f(x)e in π l x dx 1 f(x)e in π l x dx = l f ( n π ). l f(x) n= f ( n π ) e in π l x π l l (x [ l, l]). l f(ξ)e iξx dξ f ( ) (2.3) f(x) = f(ξ)e iξx dξ (x R) g g Fourier g = F g (2.4) g(x) = F g(x) := g(ξ)e iξx dξ (x R) F : g g Fourier (2.3) Fourier (2.5) F (Ff) = f. (2.6) F (F g) = g. f Fourier Fourier ( g Fourier Fourier g ) Fourier Fourier Fourier Fourier 1 4 f [ l, l] 0 38

40 f Fourier f(ξ) = Ff, g Fourier ( Fourier ) g(x) = F g (2.7) (2.8) f(ξ) = Ff(ξ) := 1 g(x) = F g(x) := 1 f(x)e iξx dx, g(ξ)e iξx dξ (2.9) F (Ff) = f, F (F g) = g (2.10) (2.11) f(x) = 1 g(ξ) = 1 ( 1 ) f(x )e iξx dx e iξx dξ, ( 1 ) g(ξ )e ixξ dξ e iξx dx (2.9) (2.10), (2.11) Fourier (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) f(x)e iξx dx (ξ R) Fourier f(x) = 1 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 ) 5 f Fourier c n ˆf(n) 6 A = (a ij ) Hermite (a ji ) A 39

41 S ( ) F : S S F 2.2 Fourier Fourier ( ) Fourier ( Fourier ) ( ) ( ) Mathematica ( ) Fourier ( ) f(x) = e x ( ) Ff(ξ) = (2.12) F [ e x ] (ξ) = Fourier 2 1 π ξ π ξ F[( ) ](Fourier ) (2.12) F[f(x)](ξ) = Ff(ξ). F [ e x ] (y) = 2 1 π y ( Mathematica FourierTransform[] ) 40

42 2.2.3 ( ) (a) (2.13) g = Ff F g = f. (b) f Fourier, f Fourier : (2.14) F f(x) = Ff( x), Ff(ξ) = F f( ξ). (a) g = Ff F (Ff) = f f = F g F(F g) = g f = F (Ff) = F g. g = F(F g) = Ff. (b) Fourier F F g(x) = 1 F f(ξ) = 1 g(ξ)e ixξ dξ (x R) f(x)e iξx dx (ξ R) (x ξ f g ) Fourier Ff(ξ) = 1 f(x)e ixξ dx (ξ R) F f(ξ) = Ff( ξ), F f( ξ) = Ff(ξ) (a), (b) (2.15) g = Ff Fg(ξ) = f( ξ). (f Fourier g g Fourier f ) 41

43 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 ( Fourier ) a > 0 (1) F [ e a x ] (ξ) = [ (2) F 1 x 2 + a 2 (3) f(x) := sinc x := sin x x 2 π ] (ξ) = 1 a a ξ 2 + a. 2 π 2 e a ξ. 1 ( a < x < a) 2a 0 ( ) sinc(aξ) Ff(ξ) =. [ ] sin (ax) (4) F (ξ) = ax [ (5) F e ax2] (ξ) = 1 e ξ2 4a. 2a Ff(ξ) = 1 sin(aξ). aξ 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 e ay e iyξ dy 1 1 a iξ (R ). 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. 42

44 1 e a x e ixξ dx = 1 ( 1 a + iξ + 1 ) = 1 a iξ (2.16) F [ e a x ] (ξ) = (2) [ F 1 x 2 + a 2 2 a π ξ 2 + a. 2 ] (ξ) = 1 e ixξ x 2 + a 2 dx. 2a 2 ξ 2 + a = 2 π 2a ξ 2 + a 2. ( 2.2.5) (1) Ff = g Fg(ξ) = f( ξ) (2.16) [ ] 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) 8 [6] 43

45 2.2.1 (sinc ) sinc ([sínk], the sinc function, the cardinal sine function) Woodward- Davies [7] (1952 ) (2.17) sinc x := sin(πx) πx (Mathematica ) sinc x := sin x (2.17) the normalized sinc function ( x ) [7] 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) e ax2 Fourier e ax2 (a ) [ (2.18) F e ax2] (ξ) = 1 e ax2 e iξx dx = 1 e ξ2 4a 2a 44

46 Fourier ( ) a = 1/2 : [ ] F e x2 /2 (ξ) = e ξ2 /2. 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 = a x + iξ ) 2 ξ2 2a 4a F [e ax2] (ξ) = e ξ2 4a 1 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 y2 1 a dy = [ (2.19) F e ax2] (ξ) = 1 e ξ2 4a. 2a ( ) 4 X, X, X + iξ 2a Cauchy 0 = = = C e az2 dz e az2 dz + [ X,X] X X e ax2 dx + π a. 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 e az2 [ X+i ξ, X] 2a dz. dz 9 m, σ 2 1 ) exp ( (x m)2 σ 2σ m = 0, σ =

47 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 e ξ 2 4a e ax 2 2a (z [ X, X + i ξ ] ). 2a [X,X+i 2a] e az2 ξ dz [X,X+i 2a] ξ e az2 ξ dz e 2 4a e ax 2 ξ 2a 0 ( X ). ( ) g(ξ) := 1 e ax2 e ixξ dx 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) ) 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 46

48 2.2.5 : 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 1 0 p > 0 ) (2.20) e iξx πe a ξ dx = x 2 + a2 a (ξ R) ( ξ ξ = 0 ) (2.20) 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 ξ > 0 e ixξ x 2 + a dx = e iξy πea( ξ) dy = 2 y 2 + a2 a = 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] 47

49 Fourier ( ) Mathematica ( N ) Mathematica Fourier FourierParameters->{0,-1} 10 f[x] Fourier ( y ) 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 Mathematica 1 f(x)e ixξ dx ixy ixy FourierParameters->{0,-1} 11 Fourier Fourier 48

50 2.3.1 (2.21) (2.22) F(f 1 + f 2 ) = Ff 1 + Ff 2, F(λf) = λff Fourier Fourier (2.23) Ff(ξ) = F f( ξ). F g(x) = 1 F f(ξ) = 1 F f( ξ) = 1 g(ξ)e ixξ dξ f(x)e iξx dx. f(x)e iξx dx = Ff(ξ) (2.24) (2.25) 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 (2.26) 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 49 f(y)e i y a ξ 1 a dy ( ) ξ. a

51 2.3.5 Fourier (2.27) F [f (x)] (ξ) = (iξ)ff(ξ). ( k N) F [ f (k) (x) ] (ξ) = (iξ) k Ff(ξ). 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 (2.28) d Ff(ξ) = if [xf(x)] (ξ). dξ ( k N) d dξ Ff(ξ) = d dξ = 1 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 : 1 ( ) ( ) : f u (2.29) (2.30) 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 50

52 u(x, t) x Fourier û(ξ, t) : û(ξ, t) = F [u(x, t)] (ξ) = 1 u t = u xx Fourier u(x, t)e ixξ dx (ξ R, t > 0). 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) tû(ξ, t) = ξ2 û(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 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 (2.31) 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. (2.29), (2.30) u ( ) 51

53 G (fundamental solution), Green (Green function), (heat kernel) G t > 0 G(x, t) > 0, G(x, t) dx = 1 G 0, 2t 12 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 lim u(x, t) = f(x) ( ) t m, σ 2 N(m; σ 2 ) 1 ) ( σ 2 exp (x m)2 2σ 2 52

54 ( (2.30) ) ( ) G Dirac : 2 G(x, t) = G(x, t), t x2 G(x, 0) = δ(x). 0 ( ) ( ) 7. ψ : R R (2.32) (2.33) 2 u t (x, t) = 2 u (x, t) 2 x2 u(x, 0) = 0, ((x, t) R (0, )), u (x, 0) = ψ(x) (x R) t u ( ) (1) u x Fourier û(ξ, t) = 1 u(x, t)e ixξ dx (2) û Fourier u ( ( G.1 ) u(x, t) = 1 2 ) x+t x t ψ(y) dy 2.5 ( ) Fourier Fourier Fourier Fourier 2.6 : ( ) ( ) Fourier Fourier ( ) (1) f R Ff Ff f 1. f = sup f(x), f 1 = x R f(x) dx. 13 f Ff f Ff 53

55 (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 [8] (Riemann ) [9] (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 (iii) f(x) = e ax2 C x ± Ff(ξ) = e ξ2 4a ξ ± Ff C ξff(ξ) 1 16 sin(aξ) ξ ξ sin(aξ) ξ ( ) 54

56 ( ) : 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 N 1 C N 1 Fourier W = 1 1 ω 1 ω 1 2 ω 1 (N 1) N 1 ω 2 1 ω 2 2 ω 2 (N 1)... 1 ω (N 1) 1 ω (N 1) 2 ω (N 1)(N 1) (W (n, j) 1 N ω (n 1)(j 1) ) ω = e i/n ω 1 ω 1 2 ω 1 (N 1) W 1 = 1 ω 2 1 ω 2 2 ω 2 (N 1)... 1 ω (N 1) 1 ω (N 1) 2 ω (N 1)(N 1) (W 1 (j, n) ω (j 1)(n 1) ) U := NW unitary ( Fourier ( ) unitary ) 56

58 3.1 Fourier ( T, nx 2nπx/T T ) f : R C ( ) (3.1) c n := 1 0 f(x)e inx dx (n Z) ( [, π] [0, ] ) (3.2) f(x) = c n e inx. 1 [0, ] N n= (3.3) h := N, x j = jh (j Z) x j f j := f (x j ) (j Z) 1 {f j } N 1 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 N ( F (t j 1 ) + F (t j ) F T N := h = h (t0 ) + F (t 1 ) + + F (t N 1 ) + F (t ) N), h := b a N j=1 F b a F (t 0 ) = F (a) = F (b) = F (t N ) : N 1 T N = h j=0 F (t j ) ( ). 1 f {f j } n Z f j+n = f j (j Z) N {f j } j Z N {f j } N 1 j=0 2 57

59 (3.1) C n : C n := 1 N 1 h f(x j )e inx j. (3.4) ω := e i/n = e ih j=0 e inx j (3.5) C n = 1 N f Fourier = e injh = ω nj C n = 1 N 1 f j ω nj. N N 1 j= (1 N ) N N j=0 f j ω nj. 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 { (1) m Z N 1 j=0 ω mj = N (m 0 (mod N)) 0 ( ). ω m = e i N m m 0 (mod N) ω m = 1 1 m N 1 m ω m 1, ω N = 1 (2) m 0 (mod N) ω m = 1 j ω mj = 1. N 1 j=0 1 = N. ω m 1 N 1 j=0 ω mj = N 1 j=0 (ω m ) j = 1 (ωm ) N 1 ω = 1 ( ) ω N m m 1 ω = 0. m 58 N 1 j=0 ω mj =

60 2 l, m l m (mod N) l m ( Fourier ) f : R C N N h := N, ω := ei/n = e ih, 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 (3.6) 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 c n e inx f j = f(x j ) = N 1 j=0 n= f j ω (n+n)j = 1 N c n e inx j = n= N 1 j=0 c n e injh = f j ω nj = c n. n= c n ω nj. C n = 1 N = 1 N N 1 j=0 m= f j ω nj = 1 N N 1 c m j=0 N 1 j=0 ( m= c m ω mj ) ω nj ω (m n)j = 1 c m N = c m. N m n m n (1) {C n } n Z N {C n } N 1 n=0 ( ) 59

61 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 1,N = c 1 + c 1+N + c 1 N + c 1+2N + c 1 2N + c 1, c N 1, A C N 1 c 1 c N 1 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 3.2 Fourier N 60

62 f = C 0 C 1. C N 1 f 0 f 1. f N 1 CN Fourier C n = 1 N N 1 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 1) N ω 0 ω 2 ω 4 ω 2(N 1), ω 0 ω (N 1) ω (N 1)2 ω (N 1)(N 1) f = f 0 f 1., C = C 0 C 1. f N 1 C N 1 (3.7) C n = 1 N N 1 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 N 1 W 1 = ω 0 ω 2 ω 4 ω 2(N 1) ω 0 ω N 1 ω (N 1)2 ω (N 1)(N 1) C n = 1 N N 1 j=0 f j ω nj (n = 0, 1,, N 1) f j = N 1 n=0 ω jn C n (j = 0,..., N 1). ( 0 W (n, j) 1 N ω nj, W 1 (j, n) ω jn ) 61

63 (3.7) 0 (N 1 ) W (n, j) 1 N ω nj W (j, k) ω jk (ω jk ) (n, k) { { N 1 1 N ω nj ω jk = 1 N 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 = N 1 c m ω mj = n=0 m n N 1 c m ω nj = n=0 m n N 1 n=0 ω nj m n c m = N 1 n=0 N 1 n=0 c n ω nj ω nj C n W 1 ω jn W f (W unitary ) U := NW (3.8) 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 n Z 62

64 3.2.2 ( 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 N 1 j=0 f j ω nj, f j = N 1 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 N 1, φ n := f = N 1 n=0 c n φ n ω n 0 ω n 1 ω n 2. ω n (N 1) C N {φ n } f (φ n, φ m ) = N 1 j=0 ω nj ω mj = N 1 j=0 ω nj ω mj = N 1 j=0 ω (n m)j = { N (n = m) 0 (n m) ( ) {φ n } f = N 1 n=0 C nφ n C n = (f, φ n) (φ n, φ n ) = N 1 j=0 f jω nj N = 1 N N 1 j=0 f j ω nj. 63

65 3.2.4 ( 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 1 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 1 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 C n = c m = c n + (c n+pn + c n pn ). m n p=1 0 n M n + pn N > 2M > M c n+pn = 0. n pn M N < M c n pn = 0. C n = c n. 3.3 FFT (the fast Fourier transform) FFT ( [10] ) FFT Fourier N N 2 N = 2 m (m ) ( ) O(N log N) Fourier FFT N CD 44.1 khz = ( ) = ( ) Fourier Fourier Fourier Fourier 3.4 Fourier ( 2016 ) 65

67 3.4.1 {C n } f : R C N Fourier (3.9) h = N, x j = jh, f j = f(x j ), ω = e ih, C n = 1 N 1 f j ω nj. N j=0 N S N (x) (3.10) S N (x) := N 2 1 k= N 2 N 1 2 k= N 1 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 (3.10) 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) 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 n k= n c k e ikx f(x) sin(kx)dx Fourier C k Fourier ( ) s n (x) Fourier c k C k S n (x) = n k= n C k e ikx ( ) 2n + 1 Fourier C k ( k n) N 2n + 1 N = 2n + 1 N N ( N = 2 m ) (3.11) S N (x) := Ck e ikx, k N/2 k N 2 66

68 (a) N k N/2 k k = N 1, N 1 2 1,..., N 1 ( N ) 2 Ck e ikx := k N 2 k N/2 C k e ikx = k= N 1 2 k= N 1 2 C k e ikx. (b) N N/2 k N/2 k k = N/2, N/2+1,, N/2 N + 1 (1 ) 3 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 (N 1)/2 C k ω kj k= (N 1)/2 C k ω kj (N ) (N ) = N 1 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 ) = N 1 k=0 C k ω kj = f j = f(x j ) x j x R S N (x) 1 x S N (x) (3.10) x j+1/2 = x j + h/2 (j = 0, 1,, N 1) (Fourier (2.25) ) 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) 3 k N 2 Ck e ikx := N/2 k= N/2+1 C k e ikx 2 67

69 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) 3.5 Mathematica Fourier Mathematica Fourier[f ] f = {f 0, f 1,..., f N 1 } N 1 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 N 1 c ij = L l, l = (j i) mod N L 0 L 1 L N 2 L N 1 L N 1 L 0 L 1 L N 2 C = L 2... L 0 L 1 L 1 L 2 L N 1 L 0 U := 1 N (ω nj ) φ 0 0 U φ 1 CU = diag [φ 0, φ 1,, φ N 1 ] =..., φ p := 0 φ N 1 68 N 1 j=0 ω pj L j

70 C C 1, det C ( ) 3.7 : ( ) 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 : (3.12) A n := 2 N N 1 j=0 f j cos(nx j ), B n := 2 N N 1 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. 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 (N 1)/2, B (N 1)/2 (N f B N/2 = 0 ) N Fourier 69

71 N R N (f 0, f 1,, f N 1 ) (A 0, A 1, B 1,, A N 1, B N 1, A N ), A n = 2 N B n = 2 N N 1 j=1 N 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 N 1 ) A 0, A 1, B 1,, A N 1 2, B N 1 2 ), A n = 2 N B n = 2 N N 1 j=0 N 1 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 (N 1)/2 n=1 ( A n cos nj N + B n sin nj ) N (j = 0, 1,, N 1). (C N int N N/2 N 1 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 (3.13) A n = 1 N ( f N 1 j=1 f j cos πnj N + ( 1)n f N ) (n = 0, 1,, N) (f 0, f 1,, f N ) (A 0, A 1,, A N ) 70

72 R N+1 R N+1 ( ) (3.14) f j = 1 N 1 A A n cos πnj 2 N + ( 1)j A N n=1 (j = 0, 1,..., N). (3.14) 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 N 1 j=0 N 1 j=1 N 1 j= f j = 2N 1 j=0 N 1 j=1 C n ω nj = C 0 + f j ω nj f j ω nj + f N ω nn + N 1 j=1 f j ( ω nj + ω nj) + ( 1) n f N ) f j cos πnj N N 1 n=1 + ( 1)n f N C n ω nj + C N + ) f 2N j ω n(2n j) ). N 1 n=1 ( ) = 1 N 1 ( A 0 + An ω nj + A n ω nj) + ( 1) j A N 2 n=1 ( ) = 1 N 1 A A n cos πnj 2 N + ( 1)j A N. n=1 C 2N n ω (2N n)j 71

73 3.7.4 (f 1,, f N 1 ) R N 1 (3.15) B n = 2 N N 1 j=1 f j sin πnj N (n = 1, 2,, N 1) (f 1, f 2,, f N 1 ) (B 1, B 2,, B N 1 ) R N 1 R N 1 (3.16) f j = N 1 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 N 1 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 ( N 1 j=1 N 1 j=1 N 1 j=1 2N 1 j=0 f j ω nj + f j ω nj N 1 j=1 f j ( ω nj ω nj) f j sin πnj N. f 2N j ω n(2n j) ) 72

74 3.2.1 f j = = 1 2i 2N 1 j=0 N 1 n=1 C n ω nj = C 0 + N 1 n=1 ( Bn ω nj B n ω nj) N 1 = B n sin πnj N. n=1 C n ω nj + C N + N 1 n=1 C 2N n ω (2N n)j 73

75 4 ( Mathematica ) (1) WWW 1 WAVE guitar-5-3.wav 2 ( ) 3 (2) Mathematica (1) ( ) SetDirectory["~/Desktop"] FileNames[] guitar-5-3.wav Fourier ( [11]) snd=import["guitar-5-3.wav"] 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[" mk/fourier/guitar-5-3.wav"] URL Import 3 Safari control 74

76 tb = Take[tbl, {1, 3*sr}]; g = ListPlot[tb, PlotRange -> All] sr 3 ( sr = 44.1 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]]; ( ) 75

77 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 ( ( 5.0.1) 3.2.4(2016/11/16 ) ) 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 76

78 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 N 1 j=0 x j ω nj (n Z), ω = e i/n. {C n } N {C n } N 1 n=0 {C n } N 1 n=0 C n = {C n } {x j } Fourier : (4.4) x j = N 1 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 77

79 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.. C 129 = C N 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) = sin nπx L L a n = 2 L 0 ( ) n 2L nc n=1 ϕ(x) sin nπx L dx, b n = 2 cnπ ( a n cos cnπt L + b n sin cnπt ), L L 0 ψ(x) sin nπx L nc,. (n = 1 ) c 2L 78 dx. 2L

80 4.3.1 ( ) (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 A n := (f n/t ). T 0 e i(f n/t )t dt = 1 T c n = sinc A nt 2. T 0 e iant dt = 1 ia n T ( e ia nt 1 ). T = 1 s, f = Hz n = 125,, 135 sinc(a n T/2) ( ) 4.4 Mathematica Mathematica Fourier[ ] C n = 1 N 1 x j ω nj N 79 j=0

81 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

82 5 ( ) Fourier ( 3.2.4) f : R C [12] ( ) ( ) 1 (Harry Nyquist, , ) (Calude E. Shannon, ) [4] 3 (1949 ) [13] (1949 ) Kotel nikov 1933 E. T. Whittaker ( , ) 1915 ( , ) 1920 (Butzer [14] Whittaker Whitakker ) 1 ( ) 2 Certain topics in telegraph transmission theory W 1 2W 3 Communication in the presence of noise 81

83 5.0.1 (, 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= 82

84 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 ). ( ) 83

85 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. 84

86 {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= N 1 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 1 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 85