1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier

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Fourier Fourier Fourier etc * 1 Fourier Fourier Fourier (DFT Fourier (FFT Heat Equation, Fourier Series, Fourier Transform, Discrete Fourier Transform, etc Yoshifumi TAKEDA 1 Abstract Suppose that u is a function describing the temperature at a location in a metal bar The heat equation is a parabolic partial differential equation which is used to determine the change in the function u over time We explain the solution technique called separation of variables, which is a traditional method In this article, we consider it especially in ideal conditions, in order to give an explanation of the principles of Fourier series and Fourier transform And then we shall elaborate on discrete Fourier transform (DFT and fast Fourier transform (FFT from a mathematical point of view *1 Department of Mathematics and Statistics, Wakayama Medical University 1

1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier Fourier

Fourier Fourier 014 1 u(x,t x t u u x t 3

1 u t u x (1 * 3 (1 *3 u(x,t g(x h(t u(x,t g(x h(t u t g(x d dt h(t, u x d g(x h(t dx g(x d dt d h(t g(x h(t dx h (t h(t g (x g(x x t x t *4 h (t h(t g (x g(x ξ ( h (t ξ h(t, g (x ξ g(x h(t e ξ t g(x e ξ ix, e ξ ix * *3 Fourier Jean Baptiste Joseph Fourier (1768-1830 *4 ξ 4

(1 e ξ t e ξ ix, e ξt e ξ ix Ae ξ t e ξ ix + Be ξ t e ξ ix (A + Be ξ t cos( ξ x + (A Bie ξ t sin( ξ x ξ A, B e ξ t cos( ξ x, e ξ t sin( ξ x (3 4 Fourier t 0 5

(1 (3 t 0 u(x,t Ae t cosx *5 t 0 *6 Fourier π π π π *5 A t 0 *6 ( ξ 6

π 0 π n1 n 1 e (n 1t sin((n 1x u(x,t (t 0 u(x,0 n1 sin((n 1x n 1 sinx + 3 sin(3x + 5 sin(5x + 7 sin(7x + n 10 n 100 7

Fourier * 7 *8 Fourier (t 0 f (x f (x u(x,t x cos(nx, sin(nx n e n t u(x,t ( e n t A n cos(nx + B n sin(nx n1 (4 Fourier * 9 A n, B n f (x u(x,0 n1 ( A n cos(nx + B n sin(nx (5 (4 f (x (5 A n, B n *7 *8 *9 0 8

π π π π π π π π cos(mx sin(nxdx 0 cos(mx cos(nxdx 0 (m n sin(mx sin(nxdx 0 (m n cos (nxdx π π sin (nxdx π π f (x cos(mxdx π π π ( n1 n1 ( A n π ( A n cos(nx + B n sin(nx cos(mxdx π cos(nx cos(mxdx + B n π π A n cos(mx cos(mx dx A n π π π sin(nx cos(mx dx (6 sin(mx A n 1 π π π f (x cos(nxdx, B n 1 π π π f (x sin(nxdx A n, B n f (x (5 (4 (5 Fourier A n, B n Fourier 5 Fourier e nt cos(nx, e nt sin(nx 9

* 10 e ξ t cos( ξ x, e ξ t sin( ξ x (3 ξ ξ ω e ωt cos(ω x, e ωt sin(ω x e ωt e iω x, e ωt e iω x ω ω e ωt e iω x (7 * 11 t 0 ω C(ω C(ωe iω x C(ωe iωx dω f (x f (x C(ωe iωx dω Fourier C(ω Fourier *10 *11 0 10

Fourier C(ω f (xe iηx dx ( * 1 C(ω 1 f (x C(ωe iωx dω e iηx dx C(η f (xe iωx dx (8 C(ωe iωx dω (9 f (x C(ω (8 Fourier C(ω f (x (9 Fourier Fourier 1 Gauß f (x e x σ Fourier * 13 C(ω 1 f (xe iωx dx 1 e x σ e iωx dx σ e σ ω Fourier C(ω Gauß f (x σ C(ω ω Fourier *1 (6 1 *13 13 11

Fourier f (x 1 T ( T < x < T 0 (x T, T x 1 T T T C(ω 1 1 4πT T T f (xe iωx dx 1 T cos(ωxdx 1 4πT 1 T e iωx dx 1 4πT [ ] sin(ωx T ω T T T T sin(t ω T ω e iωx dx f (x T sin ω ω 0 6 ω A(ω, B(ω A(ω B(ω A(ω B(ω A(η B(ω ηdη (10 1

Fourier A(ω B(ω e iωx dω ( A(η B(ω ηdη e iωx dω ( A(η B(ω ηe iωx dω dη ( A(η B(ωe i(ω+ηx dω dη (ω ω η ( ( A(ηe iη x dη B(ωe iω x dω C Fourier F (C F (A B F (A F (B Fourier 7 * 14 ω f (x f (x u(x,t e ωt e iω x (7 C(ωe iωx dω C(ωe ωt e iωx dω f (x C(ω 1 f (ye iωy dy (8 *14 13

u(x,t 1 1 K(x,t ( 1 ( ( f (ye iωy dy e ωt e iωx dω f (ye iωy e ωt e iωx dω dy e ωt+iω(x y dω f (ydy K(x,t 1 e ωt+iωx dω f (x u(x,t u(x,t K(x,t * 15 K(x y,t f (ydy K(x,t 1 πt x e 4t * 16 8 Fourier (DFT *15 Green *16 14 14

* 17 N f (x N x 1, x,, x N f (x 1, f (x,, f (x N f (x N 1 f (x N N 1 N 1 N x N 1 *17 15

* 18 N N (a 1, a,, a N N 1 c 0 + c 1 x + + c x N (c 0, c 1,, c f (x f (x c 0 + c 1 x + + c x (11 1, x,, x cos(nx, sin(nx (n 1,, 3, f (x f (x A 1 cosx + B 1 sinx + A cos(x + B sin(x + (11 Fourier (c 0, c 1,, c Fourier *18 C[x] C[x]/(x N 1 16

Fourier Fourier * 19 Fourier ( Fourier ( Fourier ( Fourier ( 1 N 1, e i N 1, e i N i,, e N N e i 1 N ζ 1, ζ, ζ,, ζ * 0 f (x c 0 + c 1 x + + c x (c 0, c 1,, c ( f (1, f (ζ, f (ζ,, f (ζ x ζ k *19 Fourier : Discrete Fourier Transform (DFT *0 ζ N 1 x N 1 17

f (ζ k c 0 + c 1 ζ k + + c ζ k( c 0 (1, ζ k, ζ k( c 1 c Foureir f (1 f (ζ f (ζ 1 1 1 1 ζ ζ 1 ζ ζ (( c 0 c 1 c 1 1 1 1 ζ ζ 1 ζ ζ (( 1 1 1 1 ζ 1 ζ ( 1 ζ ( ζ (( N N * 1 1 N 1 1 1 1 ζ 1 ζ ( 1 ζ ( ζ (( Foureir 9 x N 1 f (x c 0 + c 1 x + + c x, g(x d 0 + d 1 x + + d x x N 1 1 1 x, x 1, x x n 1, x 3 x N,, x x 1 + ζ k k + ζ (k k + + ζ ((k k 1 + 1 + + 1 N (k l *1 (k + 1,l + 1 1 + ζ k l + ζ (k l + + ζ ((k l ζ N(k l 1 ζ k l 1k l 1 1 ζ k l 0 1 (k l 18

N c 0 d 1 + c 1 d 0 + c d + + c d 1 N 0 1 k c 0 d k + c 1 d k 1 + c d k + + c d k+1 c i d k i (1 i0 N 0 c i d i + i0 f (x c 0 + c 1 x + + c x, g(x d 0 + d 1 x + + d x c i d 1 i x + + i0 f (x g(x ( i0 k c i d i, i0 c i d 1 i,, c i d i x i0 c i d i i0 c i d k i (1 i0 x k Fourier C(η D(ω ηdη (10 ω e iωx Fourier * (10 dη (1 1 ω k * 3 i η * Fourier (9 Fourier (5 *3 R (, Z/N {0, 1,, N 1} 19

Fourier Fourier Fourier Fourier Fourier x N 1 10 Fourier (FFT c i d k i i0 N N 1 N 1 0 N 1 N N N Fourier (a 0 b 0, a 1 b 1,, a b N 0

Fourier Fourier Fourier Fourier Fourier 1 1 1 1 ζ ζ 1 ζ ζ (( N 0 0 0 N 0 N * 4 Fourier Fourier * 5 Fourier Fourier Fourier Fourier D ζ 1 1 1 1 ζ ζ D ζ 1 ζ ζ (( f (ζ 0 f (ζ 1 f (ζ D ζ c 0 c 1 c N m (ζ N 1 1 + 1 + + 1 N (if k + l 0 or N *4 (k +1,l +1 1+ζ k+l +ζ (k+l + +ζ ((k+l ζ N(k+l 1 ζ k+l 1k+l 1 1 ζ k+l 1 0 (otherwise *5 N 1

(ζ 0, (ζ 1,, (ζ N 1, (ζ N, (ζ N +1,, (ζ N + N 1 f (x c 0 + c x + + c N x ( N 1 f (x f (x c 0 + c x + + c N x N 1 f (x f (x f (1, f (ζ, f (ζ,, f (ζ f (ζ 0, f (ζ 1,, f (ζ ( N 1, f (ζ N, f (ζ ( N +1,, f (ζ ( N + N 1 η ζ η N Fourier f (1 1 1 1 f (η 1 η η N 1 f (η N 1 1 η N 1 η ( N 1( N 1 1 η c 0 c c N D η D ζ N N (N 1N N N N N N 4 N 1 1 4

Fourier Fourier 1 4 + 1 4 1 * 6 f (x c 0 + c 1 x + + c x c 0 + c x + + c N x N + x(c 1 + c 3 x + + c x N g(x c 0 + c x + + c N x N h(x c 1 + c 3 x + + c x N f (x g(x + x h(x f (ζ 0 g(η 0 + ζ 0 h(η 0 f (ζ 1 g(η 1 + ζ 1 h(η 1 f (ζ N 1 g(η N 1 + ζ N 1 h(η N 1 f (ζ N g(η N + ζ N h(η N f (ζ N +1 g(η 1 + ζ N +1 h(η 1 f (ζ g(η N 1 + ζ h(η N 1 g, h * 7 f, g, h (c 0,,c, (a 0,,a N 1, (b 0,,b N 1 Fourier *6 *7 η N 1 3

(ĉ 0,,ĉ, (â 0,,â N 1, ( b 0,, b N 1 ĉ 0 â 0 + ζ 0 b 0 ĉ 1 â 1 + ζ 1 b 1 ĉ N 1 â N 1 + ζ N 1 b N 1 ĉ N â 0 + ζ N b 0 ĉ N +1 â 1 + ζ N +1 b 1 ĉ â N 1 + ζ b N 1 ζ N 1 ĉ 0 â 0 + ζ 0 b 0 ĉ 1 â 1 + ζ 1 b 1 ĉ N 1 â N 1 + ζ N 1 b N 1 ĉ N â 0 ζ 0 b 0 ĉ N +1 â 1 ζ 1 b 1 ĉ â N 1 ζ N 1 b N 1 ( ĉ 0 â 0 ζ 0 b0 + ĉ N 1 â N 1 ζ N 1 b N 1 ĉ N ζ 0 b0 ĉ â 0 â N 1 ζ N 1 b N 1 (13 Fourier 4

(â 0,,â N 1, ( b 0,, b N 1 N N N Fourier (ĉ 0,,ĉ Fourier N N N N Fourier ( ( N N + N N + N g, h 4 * 8 1 1 Fourier Fourier 0 Fourier Fourier * 9 * 30 N ( 3 log N 1 + 1 N 1 Fourier Fourier f g â, b Fourier ĉ Fourier (N ( 3 log N 1 + 1 + N + N ( 3 log N 1 + 1 9 N log N N + 3 *8 N m *9 Fourier : Fast Fourier Transform (FFT *30 16 5

* 31 N 1 N(N + N 1 N N N 11 c c 0 + c 1 + + c, d d 0 + d 1 + + d f (x c 0 + c 1 x + + c x, g(x d 0 + d 1 x + + d x f (x, g(x Fourier f (x g(x x c d f ( g( *31 Fourier Fourier 6

1 11 Gauß Gauß e x dx π, x t, x t dt, dx, e t dx Gauß R, a R Cauchy R R a R 0 e x dx + e (R+iy idy + e (x+ia dx + e ( R+iy idy 0 0 R a a a e (R+iy idy e (R+iy a dy e R +y iry a dy e R +y dy 0 0 e R +a a dy e R +a a e R +a a 0 as R 0 a lim e (R+iy idy 0 R 0 0 0 0 lim e ( R+iy idy 0 R a e x dx e (x+ia dx (14 * 3 *3 Cauchy x + ia u, dx du e (x+ia dx e u du 7

1 Fourier f (x C(ω f (x Fourier f (xe iηx dx Gauß ( C(ωe iωx dω h(x e x C(ωe iωx dω e iηx dx h(εx e ε x 1 as ε 0 f (x h(εx f (xe iηx dx ( h(εx ( ( εx z, x z, εdx dz ε ( ω η εζ, dω εdζ ( C(ωe iωx dω e iηx dx h(εxc(ωe iωx e iηx dω dx h(εxc(ωe i(ω ηx dω dx h(zc(ωe i(ω η z dz ε dω ε h(zc(η + εζ e iζ z dζ dz ε 0, ( ( ( ( ( h(zc(ηe iζ z dζ dz h(ze iζ z dζ dz C(η e z e iζ z dζ dz C(η e z iζ z dζ dz C(η e (z iζ (iζ dζ dz C(η 8

( ( ( e (z iζ +ζ dζ dz C(η e (z iζ e ζ dζ dz C(η e (z iζ dz e ζ dζ C(η (14 ( ( e z dz e z dz e ζ dζ C(η ( e ζ dζ C(η Gauß C(η C(η C(ω 13 Fourier Gauß f (x e x σ Fourier C(ω 1 f (xe iωx dx 1 e x σ e iωx dx 1 1 1 1 1 1 9 e x σ iωx dx e x +σ iωx σ dx e (x+σ iω σ 4 i ω σ e (x+σ iω +σ 4 ω σ dx dx e (x+σ iω σ e σ4 ω σ dx e (x+σ iω σ dx e σ ω

(14 x σt, dy σdt Gauß Gauß 1 e x σ dx e σ ω 1 σ σ e t σ dt e σ ω e t dt e σ ω e σ ω σ e σ ω 14 7 K(x,t 1 1 1 x e 4t e ω t+iωx dω e (ω t ix t x 4t dω e (ω t ix t dω ω t η, t dω dη 1 t x e 4t (η ix e t dη (14 1 t x e 4t e η dη Gauß 1 t 1 πt e x 4t π x e 4t 30

15 Parseval-Plancherel Fourier c (c 0, c 1,, c, d (d 0, d 1,, d ĉ ( f (1, f (ζ, f (ζ,, f (ζ, d (g(1, g(ζ, g(ζ,, g(ζ ĉ l f (ζ l c k ζ kl k0 Fourier (1/ND ζ d Fourier d 0 d 1 d 1 N 1 1 1 1 ζ 1 ζ ( 1 ζ ( ζ (( d 0 d 1 d d k 1 N ζ kl g(ζ l l0 ( 1 (c,d c k d k c k k0 k0 N ζ kl g(ζ l l0 1 N k0 1 N (ĉ, d c k ζ kl g(ζ l 1 l0 N l0 ( 1 c k k0 N ( c k ζ kl g(ζ l 1 k0 N ζ kl g(ζ l l0 l0 f (ζ l g(ζ l Fourier Parseval-Plancherel (c,d 1 N (ĉ, d, c 1 N ĉ 31

16 Fourier N m 1 N ζ e i 1 N Fourier (13 D ζ, D η D ζ c 0 c N 1 c N c a 0 ζ 0 b 0 D η + D η a N 1 ζ N 1 b N 1 D η a 0 a N 1 ζ 0 ζ N 1 D η b 0 b N 1 N m S(m ζ 0 1 +, S(m S(m 1 + m 1 1 + m 1 + m 1 S(m 1 + 3 m 1 1 S(m 1 (S(m 1 1 + 3 m 1 S(m 1 m S(m 1 1 m 1 + 3 S(0 0 S(m 1 m S(0 1 0 + 3 m 3 m 1 S(m m ( 3 m 1 + 1 [1] Émile Picard, Leçons sur quelques équations fonctionnelles avec des applications a divers problèmes d analyse et de physique mathématique, rédigées par Eugène Blanc, Gauthier- Villars (198 (1977 3

[] ( (1953 (1958 [3] 4 (1963 [4] (1977 [5] Rudolf Lidl and Günter Pilz, Applied Abstract Algebra, Undergraduate Texts in Mathematics, Springer, nd edition (1997 [6] (003 33

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