Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K 12.1.6 Fourier ?êÂñ5 ½n 1 (Dirichlet) ¼ê f(x) ± 2π ±Ï, 1 ◦ XJ¼ê3?Ûk«mþ´Åã1w, K§ Fourier ?ê3 ê¶þÑÂñ, a0 2 + X ∞ n=1 (an cos nx + bn sin nx) = f(x + 0) + f(x − 0) 2 ; 2 ◦ XJ¼ê??ëY, 3?Ûk«mþ´Åã1w, KÙ Fourier ? êÒ3ê¶þýéÂñu f(x). 5, ùp¤¢¼ê f(x) 3k«mþÅã1w´¼êØk: , f(x) ëY këYû f 0 (x), ùk:U´ f(x) 9 f 0 (x) 1 amä:. 1/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K f(x) ´ [−π, π] þȽýéȼê, an, bn ´ f Fourier Xê, = f(x) ∼ a0 2 + X ∞ n=1 an cos nx + bn sin nx . 3 x0 :, Fourier ?êÜ©Ú Tn(x0) = a0 2 + X n k=1 ak cos kx0 + bk sin kx0 . ò Fourier XêLª ak = 1 π Z π −π f(x) cos kx dx, bk = 1 π Z π −π f(x) sin kx dx, \Ü©Ú¥, 2/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K Tn(x0) = 1 2π Z π −π f(x)dx + X n k=1 1 π Z π −π f(x) cos kx cos kx0 + sin kx sin kx0 dx = 1 π Z π −π f(x) 1 2 + X n k=1 cos k(x − x0) ! dx. P Kn(x) = 1 2 + X n k=1 cos kx, =, Kn(x) = sin (n + 1 2 )x 2 sin x 2 , x 6= 2mπ, m ∈ Z (12.1) ¡ Dirichlet ؼê, ´± 2π ±Ïó¼ê. Ún 1 (Riemann-Lebesgue) f(x) ´ [a, b] þȽýéȼê(=, e f k., K f Riemann È; e f Ã., K |f| 2ÂÈ), Kk lim λ→+∞ Z b a f(x) cos λx dx = 0, lim λ→+∞ Z b a f(x) sin λx dx = 0. 3/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K Tn(x0) = 1 π Z π −π f(x)Kn(x − x0) dx = 1 π Z x0+π x0−π f(x)Kn(x − x0) dx = 1 π Z π −π f(x + x0)Kn(x) dx = 1 π Z π 0 f(x + x0)Kn(x) dx + 1 π Z 0 −π f(x + x0)Kn(x) dx = 1 π Z π 0 f(x + x0)Kn(x) dx + 1 π Z π 0 f(−x + x0)Kn(x) dx, ¤± Tn(x0) = 1 π Z π 0 f(x0 + x) + f(x0 − x) 2 sin x 2 sin(n + 1 2 )x dx. (12.2) 4/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K ¿©ê δ, ò (11.2) ªmàÈ©©üÜ©, k Tn(x0) = 1 π Z δ 0 F(x, x0) sin(n + 1 2 )x dx + 1 π Z π δ F(x, x0) sin(n + 1 2 )x dx (12.3) Ù¥ F(x, x0) := f(x0 + x) + f(x0 − x) 2 sin x 2 3 [δ, π] þȽýéÈ. d Riemann-Lebesgue Ún(=, Ún 1) (11.3) ª mà1È© n → +∞ 4 0. Ïd f(x) Fourier ?ê3 x0 ´ ÄÂñ, ±9ÂñoÈ© 1 π Z δ 0 f(x0 + x) + f(x0 − x) 2 sin x 2 sin(n + 1 2 )x dx (12.4) k'. du (11.4) ªÈ© f 3 x0 NCk', Ïd·e¡Û Üz½n. 5/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K ½n 2 (ÛÜz½n) ¼ê f(x) ± 2π ±Ï, 3 [−π, π] þȽýé È. K f(x) Fourier ?ê3 x0 ´ÄÂñ±9Âño, f(x) 3 x0 NCk'. 5¿ â½Â, Fourier Xê f(x) 3«m [−π, π] þk'. Ïd ÛÜz½nNy Fourier ?êAO?. ?ÚkXe½n. ½n 3 (Dini ½n) ¼ê f(x) ± 2π ±Ï, 3 [−π, π] þȽýé È. éu¢ê s, - ϕ(t) = f(x0 + t) + f(x0 − t) − 2s. e3 δ > 0, ¦¼ê ϕ(t) t 3 [0, δ] þȽýéÈ, K f(x) Fourier ?ê3 x0 ?Âñu s. 6/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K y² ~¼ê 1 Fourier ?êE´ 1, ¤±3 (12.2) ¥- f = 1, 1 = 2 π Z π 0 sin (n + 1 2 )x 2 sin x 2 dx. (12.5) ò (11.2) ª~ (11.5) ª s , Tn(x0) − s = 1 π Z π 0 f(x0 + x) + f(x0 − x) − 2s 2 sin x 2 sin(n + 1 2 )x dx = 1 π Z π 0 ϕ(x) 2 sin x 2 sin(n + 1 2 )x dx. (12.6) Ïâ^ ϕ(x) x 3 [0, δ] þȽýéÈ, ¤± ϕ(x) 2 sin x 2 = ϕ(x) x · x 2 sin x 2 3 [0, δ] þȽýéÈ, Ïdd¼ê3 [0, π] þȽýéÈ. d Riemann-Lebesgue Ún, (11.6) ªmàÈ© n → +∞ ªu 0, = lim n→∞ Tn(x0) = s. 7/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K ½Â 1 f(x) 3 x0 NCk½Â. XJ3 δ > 0, L > 0 9 α > 0 ¦ |f(x0 + t) − f(x0 + 0)| 6 Ltα ; t ∈ (0, δ] |f(x0 − t) − f(x0 − 0)| 6 Ltα , t ∈ (0, δ] K¡ f(x) 3 x0 NC÷v α Lipschitz ^. d Dini ½n, ±Xe½n ½n 4 ¼ê f(x) ± 2π ±Ï, 3 [−π, π] þȽýéÈ. e f(x) 3 x0 NC÷v α Lipschitz ^, K f(x) Fourier ?ê3 x0 Âñu s = f(x0 + 0) + f(x0 − 0) 2 . 8/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K y² t ∈ (0, δ] , - ϕ(t) : = f(x0 + t) + f(x0 − t) − 2s = f(x0 + t) + f(x0 − t) − f(x0 + 0) + f(x0 − 0) = f(x0 + t) − f(x0 + 0) + f(x0 − t) − f(x0 − 0) . Kk |ϕ(x)| 6 2Ltα . ¤± ϕ(t) t 6 2L · 1 t 1−α , 0 1 , ϕ(t) t 3 [0, δ] þk. È. 0 < α < 1 , ϕ(t) t 3 [0, δ] þý éÈ. â½n 3 (Dini ½n) , f(x) Fourier ?ê3 x0 Âñu f(x0 + 0) + f(x0 − 0) 2 . 9/16 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n Riemann-Lebesgue Ún Dini ½n ¯K ½n 5 ¼ê f(x) ± 2π ±Ï, 3 [−π, π] þȽýéÈ. e f(x) 3 x0 , ½3 x0 kkê f 0 −(x0) Úm f 0 + (x0), K f(x) Fourier ?ê3 x0 Âñu f(x0). e f(x) 3 x0 =kük2Â!mê: lim t→0+ f(x0 − t) − f(x0 − 0) −t , lim t→0+ f(x0 + t) − f(x0 + 0) t , K f(x) Fourier ?ê3 x0 Âñu s = f(x0 + 0) + f(x0 − 0) 2 . y² e f(x) 3 x0 kük2ÂêÚ2Âmê, K f(x) 3 x0 NC÷v 1 Lipschitz ^. Ïd(ؤá. 10/16 kJ Ik J I £ ¶ '4 òÑ