Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin

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1 ( ) Fourier f : R C () (2) f(x) = a (a n cos nx + b n sin nx), n= a n = f(x) cos nx dx, b n = π π f(x) sin nx dx a n, b n f Fourier, (3) f Fourier or No. ) 5, Fourier (3) (4) f(x) = c n = n= c n e inx f(x)e inx dx. c n a n, b n ( 9) 2 (a n ib n ) (n > 0) c n = 2 a 0 (n = 0) 2 (a n + ib n ) (n < 0). f ( n Z) c n = c n. ( C N n = C n C n )

2 Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin nx). n= n= x : f(x + 0) + f(x 0) 2 = a (a n cos nx + b n sin nx). n= ( Gibbs ) (b) C : ( N lim sup N f(x) a (a n cos nx + b n sin nx)) = 0. x [,π] (c) 2 ( ) (2 ) : ( N lim N f a (a n cos nx + b n sin nx)) = 0. n= ( (c) ) f(x) 2 dx < (, π) 2 n= ( OK, [, π] OK) 2 L 2 (, π) φ, ψ L 2 (, π) (φ, ψ) = φ = (φ, φ) = φ(x)ψ(x)dx φ(x) 2 dx. Fourier {φ n } n m (φ n, φ m ) = 0. 2

3 {φ n } f = n ( : ) ( : Fourier a n = π ) {φ n } f c n φ n c n = (f, φ n) (φ n, φ n ), b n = π, c n = f = n c n φ n Fourier (i) {cos nx} n 0 {sin nx} n N (ii) {e inx } n Z ( ) {φ n } def. (φ n, φ m ) = δ nm = { (n = m) 0 (n m) ( ) {φ n } 0 ψ n := φ n φ n {ψ n } (a) (b), π cos nx, π sin nx (n N) e inx (n Z) ( φ n (x) = e inx (n Z) φ n = ) {φ n } f = n c n φ n c n = (f, φ n ) {φ n } Bessel (f, φ n ) 2 f 2 {φ n } Parseval (f, φ n ) 2 = f 2 n n 3

4 ( ) ( ) c n = f(n) f (n) = f (x)e inx dx = f(x) ( in)e inx dx = in f(n). ( Fourier Fourier ( ) ) 2 Fourier f : R C Fourier Ff(ξ) = f(ξ) := Ff = f g : R C Fourier ( Fourier ) F g(x) = g(x) := F g = g f(x)e ixξ dx (ξ R) g(ξ)e ixξ dξ (x R) ( ) Fourier Fourier F Ff = f, FF g = g. Fourier Fourier f(x) = Ff(ξ) = F f( ξ), F f(x) = Ff( x). f(ξ)e ixξ dξ Ff = g F g = f ( ) Fourier (Fourier ) 3 ( 5 ) ( 3 ) a > 0 4

5 (i) F [ e a x ] (ξ) = 2 a π ξ 2 + a. 2 (ii) [ ] ( F 2 a (x) = e a x Ff(ξ) = π ξ 2 +a 2 F f( ξ) ) [ F x 2 + a 2 ] (ξ) = π a 2 e a ξ. ( ) f(x) = { ( x < a) 0 ( ) [ sin(ax) F x Ff(ξ) = 2 π ] π (ξ) = 2 sin(aξ). ξ 0 ( x > a) ( x < a) 2 (x = ±a) (x = ±a ) ( x < a) f(x) = 2a 0 ( ) Ff(ξ) = sin(aξ) aξ F [sinc(ax)] (ξ) = = sinc(aξ), 0 ( x > a) ( x < a) 2a ( x = a) 4a sinc sinc x = sin x ( ) x (iii) Gaussian Fourier : ( ) F [ e ax2] (ξ) = 2a e ξ2 4a. 5

6 Fourier ( ) F [f + f 2 ] = Ff + Ff 2. F [λf] = λff. F [f(x a)] (ξ) = e iaξ Ff(ξ). F [ f(x)e iax] (ξ) = Ff(ξ a). F [f(ax)] (ξ) = ( ) ξ a Ff. a F [f (x)] (ξ) = (iξ)ff(ξ). d Ff(ξ) = if [xf(x)] (ξ). dξ ( Fourier ) 3 Fourier T f : R C Fourier c n = T T/2 T/2 f(x)e inx/t dx = T T 0 f(x)e inx/t dx (n Z) h := T N, ω := ei/n = e ih, x j = jh, f j = f(x j ) ( ) {f j } N c n C n = N f (N ) Fourier N f j ω nj ω : N, i.e. ω N =, m N ω m. { N ω mj N (m 0 (mod N)) = 0 ( ). 6

7 {C n } N n Z C n = c k = c n + (c n+pn + c n pn ). k n p= n (N ) C n c n C N c N c f = f 0 f. f N C = CN C n = N C 0 C. N CN f Fourier ω nj f j Fourier C N C N f C C N Fourier W = N ( ω (n )(j ) ) W = ( ω (j )(n )). C n = N N ω nj f j (n = 0,,, N ) f j = N n=0 ω jn C n (j = 0,,, N ). U := NW : U U = UU = I. ( : Fourier ) N Fourier Fourier (fast Fourier transform, FFT) 4 Fourier ( ) {f n } n Z C Z f : Z C Ff(ω) = f(ω) := f(n)e inω (ω R) n= f : R C f Fourier f f(n) = f(ω)e inw dω 7 (n Z).

8 5 ( ) Fourier f : R C 5. (, Nyquist, Shannon, ) x: R C Fourier X(ω) = x(t)e iωt dt W > 0 ( ω R : ω W ) X(ω) = 0 W T := π W sin π(n t/t ) ( t R) x(t) = x(nt ). π(n t/t ) n= W 2 f s := T = W π f 2f Fourier 4B 2 6 Fourier ( ) Fourier Fourier (Fourier ) Fourier Fourier 8

9 R Fourier f(ξ) = f(x)e ixξ dx (ξ R) f(x) = R Fourier c n = T f(x)e inx dx 0 (n Z) f(x) = Z ( ) Z ( ) Fourier f(ω) = Fourier C n = n= N ω := exp f(n)e inω (ω [0, ]) f(n) = f j ω nj (0 n N ), f j = N ( ) i N n= N n=0 0 c n e inx C n ω nj f(ξ)e ixξ dξ f(ω)e inω dω (c n f(n) f j f(j), C n f(n) N ) ( ) L 2 (R) L 2 (R), L 2 (0, ) l 2 (Z), l 2 (Z) L 2 (0, ), C N C N 7 (, ) (, convolution) f g (i) f, g : R C f g : R C (5) f g(x) := f(x y)g(y) dy (x R) (ii) f, g : R C f g : R C (6) f g(x) := f(x y)g(y) dy (x R) (iii) f, g : Z C f g : Z C (7) f g(n) := k= f(n k)g(k) (n Z) 9

10 (iv) N f, g : Z C f g : Z C (8) f g(n) := N k=0 f(n k)g(k) (n Z) N (9) (0) () (2) (3) f g = g f, ( ) (f g) h = f (g h), ( ) (f + f 2 ) g = f g + f 2 g, ( ) (cf) g = c(f g) ( ) f g = 0 f = 0 or g = 0. ( ) ( ( ) ) Fourier Fourier (i) f : R C Fourier Ff(ξ) := F[f g](ξ) = Ff(ξ)Fg(ξ). (ii) f : R C Fourier Ff(n) := F[f g](n) = Ff(n)Fg(n). (iii) f : Z C Fourier Ff(ξ) := F[f g](ξ) = Ff(ξ)Fg(ξ). f(x)e ixξ dx n= (iv) N f : Z C Fourier Ff(n) := N F[f g](n) = NFf(n)Fg(n). f(x)e inx dx f(n)e inξ N f(j)ω nj (i), (ii), (iii) δ, f f δ = f 0

11 δ ( ) δ = {δ n } n Z := {δ n0 } n Z = {, 0, 0,, 0, 0, } (δ 0 =, n 0 δ n = 0). Fourier Fδ =, F = δ. ( ) 8 ( )

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