,

Size: px
Start display at page:

Download ","

Transcription

1 ,

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

,

, 2014 10 3, 2017 1 28 ( ) 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@

More information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

基礎数学I

基礎数学I I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............

More information

A 2008 10 (2010 4 ) 1 1 1.1................................. 1 1.2..................................... 1 1.3............................ 3 1.3.1............................. 3 1.3.2..................................

More information

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

Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a 0 2 + (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin ( ) 205 6 Fourier f : R C () (2) f(x) = a 0 2 + (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=

More information

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1) 1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y

More information

Microsoft Word - 信号処理3.doc

Microsoft Word - 信号処理3.doc Junji OHTSUBO 2012 FFT FFT SN sin cos x v ψ(x,t) = f (x vt) (1.1) t=0 (1.1) ψ(x,t) = A 0 cos{k(x vt) + φ} = A 0 cos(kx ωt + φ) (1.2) A 0 v=ω/k φ ω k 1.3 (1.2) (1.2) (1.2) (1.1) 1.1 c c = a + ib, a = Re[c],

More information

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

2014 3 10 5 1 5 1.1..................................... 5 2 6 2.1.................................... 6 2.2 Z........................................ 6 2.3.................................. 6 2.3.1..................

More information

1 yousuke.itoh/lecture-notes.html [0, π) f(x) = x π 2. [0, π) f(x) = x 2π 3. [0, π) f(x) = x 2π 1.2. Euler α

1   yousuke.itoh/lecture-notes.html [0, π) f(x) = x π 2. [0, π) f(x) = x 2π 3. [0, π) f(x) = x 2π 1.2. Euler α 1 http://sasuke.hep.osaka-cu.ac.jp/ yousuke.itoh/lecture-notes.html 1.1. 1. [, π) f(x) = x π 2. [, π) f(x) = x 2π 3. [, π) f(x) = x 2π 1.2. Euler dx = 2π, cos mxdx =, sin mxdx =, cos nx cos mxdx = πδ mn,

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59

More information

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z

More information

2011de.dvi

2011de.dvi 211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37

More information

確率論と統計学の資料

確率論と統計学の資料 5 June 015 ii........................ 1 1 1.1...................... 1 1........................... 3 1.3... 4 6.1........................... 6................... 7 ii ii.3.................. 8.4..........................

More information

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z

More information

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh

More information

1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.

1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1. Section Title Pages Id 1 3 7239 2 4 7239 3 10 7239 4 8 7244 5 13 7276 6 14 7338 7 8 7338 8 7 7445 9 11 7580 10 10 7590 11 8 7580 12 6 7395 13 z 11 7746 14 13 7753 15 7 7859 16 8 7942 17 8 Id URL http://km.int.oyo.co.jp/showdocumentdetailspage.jsp?documentid=

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4 1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

30 I .............................................2........................................3................................................4.......................................... 2.5..........................................

More information

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345

More information

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)

More information

phs.dvi

phs.dvi 483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....

More information

I , : ~/math/functional-analysis/functional-analysis-1.tex

I , : ~/math/functional-analysis/functional-analysis-1.tex I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................

More information

B2 ( 19 ) Lebesgue ( ) ( ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercia

B2 ( 19 ) Lebesgue ( ) ( ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercia B2 ( 19) Lebesgue ( ) ( 19 7 12 ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purposes. i Riemann f n : [0, 1] R 1, x = k (1 m

More information

newmain.dvi

newmain.dvi 数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published

More information

2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p

2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p 2012 IA 8 I 1 10 10 29 1. [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5.

More information

ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n

ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n [ ]. A = IC X n 3 expx = E + expta t : n! n=. fx π x π. { π x < fx = x π fx F k F k = π 9 s9 fxe ikx dx, i =. F k. { x x fx = x >.3 ft = cosωt F s = s4 e st ftdt., e, s. s = c + iφ., i, c, φ., Gφ = lim

More information

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou (Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.

More information

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f 22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )

More information

b3e2003.dvi

b3e2003.dvi 15 II 5 5.1 (1) p, q p = (x + 2y, xy, 1), q = (x 2 + 3y 2, xyz, ) (i) p rotq (ii) p gradq D (2) a, b rot(a b) div [11, p.75] (3) (i) f f grad f = 1 2 grad( f 2) (ii) f f gradf 1 2 grad ( f 2) rotf 5.2

More information

text.dvi

text.dvi 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

More information

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

More information

1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b

1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b 1 Introduction 2 2.1 2.2 2.3 3 3.1 3.2 σ- 4 4.1 4.2 5 5.1 5.2 5.3 6 7 8. Fubini,,. 1 1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)?

More information

Note.tex 2008/09/19( )

Note.tex 2008/09/19( ) 1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................

More information

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

More information

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a % 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2007.11.5 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory

More information

構造と連続体の力学基礎

構造と連続体の力学基礎 II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton

More information

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0. () 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >

More information

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

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

More information

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35

More information

基礎から学ぶトラヒック理論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

基礎から学ぶトラヒック理論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 初版 1 刷発行時のものです. 基礎から学ぶトラヒック理論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/085221 このサンプルページの内容は, 初版 1 刷発行時のものです. i +α 3 1 2 4 5 1 2 ii 3 4 5 6 7 8 9 9.3 2014 6 iii 1 1 2 5 2.1 5 2.2 7

More information

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =

More information

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt

More information

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a = II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [

More information

30

30 3 ............................................2 2...........................................2....................................2.2...................................2.3..............................

More information

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ

More information

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a = [ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =

More information

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2 filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin

More information

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

More information

2 2 L 5 2. L L L L k.....

2 2 L 5 2. L L L L k..... L 528 206 2 9 2 2 L 5 2. L........................... 5 2.2 L................................... 7 2............................... 9. L..................2 L k........................ 2 4 I 5 4. I...................................

More information

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V

More information

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( 5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx

More information

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

More information

Lebesgue Fubini L p Banach, Hilbert Höld

Lebesgue Fubini L p Banach, Hilbert Höld II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

More information

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3 2 2 1 5 5 Schrödinger i u t + u = λ u 2 u. u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q

More information

17 3 31 1 1 3 2 5 3 9 4 10 5 15 6 21 7 29 8 31 9 35 10 38 11 41 12 43 13 46 14 48 2 15 Radon CT 49 16 50 17 53 A 55 1 (oscillation phenomena) e iθ = cos θ + i sin θ, cos θ = eiθ + e iθ 2, sin θ = eiθ e

More information

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....

More information

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c)   yoshioka/education-09.html pdf 1 2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h

More information

4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3..........................

More information

IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x

, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x 1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d

More information

body.dvi

body.dvi ..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1 1 5 1.1........................... 5 1.......................... 14 1.3...........................

More information

1 R n (x (k) = (x (k) 1,, x(k) n )) k 1 lim k,l x(k) x (l) = 0 (x (k) ) 1.1. (i) R n U U, r > 0, r () U (ii) R n F F F (iii) R n S S S = { R n ; r > 0

1 R n (x (k) = (x (k) 1,, x(k) n )) k 1 lim k,l x(k) x (l) = 0 (x (k) ) 1.1. (i) R n U U, r > 0, r () U (ii) R n F F F (iii) R n S S S = { R n ; r > 0 III 2018 11 7 1 2 2 3 3 6 4 8 5 10 ϵ-δ http://www.mth.ngoy-u.c.jp/ ymgmi/teching/set2018.pdf http://www.mth.ngoy-u.c.jp/ ymgmi/teching/rel2018.pdf n x = (x 1,, x n ) n R n x 0 = (0,, 0) x = (x 1 ) 2 +

More information

A S hara/lectures/lectures-j.html ϵ-n 1 ϵ-n lim n a n = α n a n α 2 lim a n = 0 1 n a k n n k= ϵ

A S hara/lectures/lectures-j.html ϵ-n 1 ϵ-n lim n a n = α n a n α 2 lim a n = 0 1 n a k n n k= ϵ A S1-20 http://www2.mth.kyushu-u.c.jp/ hr/lectures/lectures-j.html 1 1 1.1 ϵ-n 1 ϵ-n lim n n = α n n α 2 lim n = 0 1 n k n n k=1 0 1.1.7 ϵ-n 1.1.1 n α n n α lim n n = α ϵ N(ϵ) n > N(ϵ) n α < ϵ (1.1.1)

More information

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F

More information

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x

More information

leb224w.dvi

leb224w.dvi 2 4 i Lebesgue Fourier 7 5 Lebesgue Walter. F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publ. Inc., New York (99) ( 49 ) 2. ( 8 ) 3. A.2 Fourier Laplace (957 ) 4. (98 ) 5. G. G. Walter, Wavelets

More information

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) = 1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,

More information

(yx4) 1887-1945 741936 50 1995 1 31 http://kenboushoten.web.fc.com/ OCR TeX 50 yx4 e-mail: yx4.aydx5@gmail.com i Jacobi 1751 1 3 Euler Fagnano 187 9 0 Abel iii 1 1...................................

More information

2 2 ( Riemann ( 2 ( ( 2 ( (.8.4 (PDF 2

2 2 ( Riemann ( 2 ( ( 2 ( (.8.4 (PDF     2 2 ( 28 8 (http://nalab.mind.meiji.ac.jp/~mk/lecture/tahensuu2/ 2 2 ( Riemann ( 2 ( ( 2 ( (.8.4 (PDF http://nalab.mind.meiji.ac.jp/~mk/lecture/tahensuu2/ http://nalab.mind.meiji.ac.jp/~mk/lecture/tahensuu/

More information

untitled

untitled 1 kaiseki1.lec(tex) 19951228 19960131;0204 14;16 26;0329; 0410;0506;22;0603-05;08;20;0707;09;11-22;24-28;30;0807;12-24;27;28; 19970104(σ,F = µ);0212( ); 0429(σ- A n ); 1221( ); 20000529;30(L p ); 20050323(

More information

, ( ) 2 (312), 3 (402) Cardano

, ( ) 2 (312), 3 (402) Cardano 214 9 21, 215 4 21 ( ) 2 (312), 3 (42) 5.1.......... 5.2......................................... 6.2.1 Cardano................................... 6.2.2 Bombelli................................... 6.2.3

More information

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)

More information

09 8 9 3 Chebyshev 5................................. 5........................................ 5.3............................. 6.4....................................... 8.4...................................

More information

Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)

Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x) 3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,

More information

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + ( IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1

More information

meiji_resume_1.PDF

meiji_resume_1.PDF β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E

More information

A

A A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................

More information

Z: Q: R: C: sin 6 5 ζ a, b

Z: Q: R: C: sin 6 5 ζ a, b Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

untitled

untitled 0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.

More information

20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33

More information

App. of Leb. Integral Theory (S. Hiraba) Lebesgue (X, F, µ) (measure space)., X, 2 X, F 2 X σ (σ-field), i.e., (1) F, (2) A F = A c F, (3)

App. of Leb. Integral Theory (S. Hiraba) Lebesgue (X, F, µ) (measure space)., X, 2 X, F 2 X σ (σ-field), i.e., (1) F, (2) A F = A c F, (3) Lebesgue (Applications of Lebesgue Integral Theory) (Seiji HIABA) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................

More information

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............

More information

Z: Q: R: C: 3. Green Cauchy

Z: Q: R: C: 3. Green Cauchy 7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................

More information

20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2 7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)

More information

I

I I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

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

i 3 Mathematica Fourier Fourier Dirac Dirac Fourier Haar 7.4 (MRA) 8 R L 2 {V j } j Z j Z V j V j+1 j Z V j = L 2 (R) f(2x) V j+1 f( Fourier 解析とウェーブレットの基礎 水谷正大著 大東文化大学経営研究所 i 3 Mathematica Fourier 1 5.4 Fourier Dirac Dirac 2.3 4 Fourier 8 6 7 Haar 7.4 (MRA) 8 R L 2 {V j } j Z j Z V j V j+1 j Z V j = L 2 (R) f(2x) V j+1 f(x) V j V 0

More information

http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................

More information

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f ,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)

More information

Z: Q: R: C:

Z: Q: R: C: 0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x

More information

prime number theorem

prime number theorem For Tutor MeBio ζ Eite by kamei MeBio 7.8.3 : Bernoulli Bernoulli 4 Bernoulli....................................................................................... 4 Bernoulli............................................................................

More information