êÚål ²²þÂñ nõª Bessel Ø ª AíØ 12.1.5 Bessel Ø ª²²þÂñ C[−π, π] ´ [−π, π] þëY¼ê N. éu f ∈ C[−π, π], - kfk0 = max −π6x6π |f(x)|. ´ kfk0 ´ C[−π, π] þê( ), = 1 ◦ kfk0 > 0, kfk0 = 0 ⇐⇒ f = 0; (5) 2 ◦ éuê λ k kλfk0 = λkfk0; (à5) 3 ◦ éu f, g ∈ C[−π, π], k kf + gk0 6 kfk0 + kgk0; (nØ ª) C[−π, π] þùê±pÑål, = d(f, g) = kf − gk0 1 ◦ d(f, g) > 0, d(f, g) = 0 ⇐⇒ f = g; (5) 2 ◦ d(f, g) = d(g, f); (é¡5) 3 ◦ éu f, g, h ∈ C[−π, π], k d(f, g) 6 d(f, h) + d(h, g); (nØ ª) 1/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ f(−π) = f(π) f 3 [−π, π] þëY ©ã1w, d Dirichlet ½ n f Fourier ?ê3 [−π, π] þÂñu f(x), = lim n→∞ kTn(x) − f(x)k0 = 0, (1) ùp Tn(x) = a0 2 + X n k=1 ak cos kx + bk sin kx (2) ´ f(x) Fourier ?êc n Ü©Ú. ¡/X Sn(x) = α0 2 + X n k=1 αk cos kx + βk sin kx (3) ¼ê n gnõª, (1) ªL«ëY ©ã1w¼êdnõ ª%C. 2/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ L2 [−π, π] ´ [−π, π] þÈ ²È¼ê N (=, éu f ∈ L2 [−π, π], e f(x) k., K f ∈ R[−π, π]. e f(x) Ã., K f Ú f 2 Ñ2 È). éu f, g ∈ L2 [−π, π], - hf, gi = 1 π Z π −π f(x)g(x) dx. K hf, gi ´ L2 [−π, π] þSÈ, - kf − gk = p hf − g, f − gi. ±y kf − gk ´ L2 [−π, π] þål. ¯K 1 éu f ∈ L2 [−π, π] ´Ä3nõª Sn(x) ¦ lim n→∞ kSn(x) − f(x)k = 0. (4) 3/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ (4) ª± lim n→∞ 1 π Z π −π Sn(x) − f(x) 2 dx = 0. (5) e (5) ¤á, K¡nõª {Sn(x)} ²²þÂñu f(x). f ∈ L2 [−π, π], Sn(x) ´d (3) ªL« n gnõª. ∆n : = 1 π Z π −π Sn(x) − f(x) 2 dx = hSn(x) − f(x), Sn(x) − f(x)i = hf, fi − 2hf, Sni + hSn, Sni. a0, a1, · · · , 9 b1, b2, · · · , ´ f(x) Fourier Xê. K hf, Sni = * f, α0 2 + X n k=1 αk cos kx + βk sin kx + = α0a0 2 + X n k=1 αkak + βkbk . 4/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ 2ân¼êX5, k hSn, Sni = * α0 2 + X n k=1 αk cos kx + βk sin kx , α0 2 + X n k=1 αk cos kx + βk sin kx + = α2 0 2 + X n k=1 α 2 k + β 2 k . u´ ∆n = hf, fi − α0a0 − 2 X n k=1 αkak + βkbk + α2 0 2 + X n k=1 α 2 k + β 2 k = hf, fi + (α0 − a0) 2 2 + X n k=1 (αk − ak) 2 + (βk − bk) 2 − a 2 0 2 − X n k=1 a 2 k + b 2 k (6) > 1 π Z π −π f 2 (x) dx − a 2 0 2 + X n k=1 a 2 k + b 2 k ! . (7) 5/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ (7) ªL²k αk = ak, βk = bk , ∆n â. Ò´`3 ¤k n gnõª¥, d f(x) Fourier Xê¤(½ Tn(x) f(x) ål. â (7) ª a 2 0 2 + X n k=1 a 2 k + b 2 k 6 1 π Z π −π f 2 (x) dx, (8) ù¡ Bessel Ø ª. ¯¢þ, k ª a 2 0 2 + X ∞ k=1 a 2 k + b 2 k = 1 π Z π −π f 2 (x) dx, (9) ù¡ Parseval ª. 6/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ â (6) ª, k kTn(x) − f(x)k 2 = 1 π Z π −π Tn(x) − f(x) 2 dx = 1 π Z π −π f 2 (x) dx − a 2 0 2 + X n k=1 a 2 k + b 2 k ! . Parseval ª`² lim n→∞ kTn(x) − f(x)k = 0, =, f(x) Fourier ?êÜ©Ú Tn(x) ²²þÂñu f(x). ùÒÑ ¯ K 1 ½5£. 7/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ ~ 1 Ï |x| = π 2 − 4 π X ∞ k=1 1 (2k − 1)2 cos(2k − 1)x, (−π 6 x 6 π). ¤±â Parseval ª, k π 2 2 + X ∞ k=1 16 (2k − 1)4π2 = 1 π Z π −π x 2 dx = 2π 2 3 . dd X ∞ k=1 1 (2k − 1)4 = π 4 96 , Xk X ∞ k=1 1 k4 = X ∞ k=1 1 (2k − 1)4 + X ∞ k=1 1 16k4 , Ï X ∞ k=1 1 k4 = π 4 90 . 8/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ íØ 1 f(x) ´ [−π, π] þÈ ²È¼ê, {an}, {bn} ´ f(x) Fourier Xê, Kk lim n→∞ an = 0, lim n→∞ bn = 0, = lim n→∞ Z π −π f(x) cos nx dx = 0, lim n→∞ Z π −π f(x) sin nx dx = 0. /, k 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. 9/17 kJ Ik J I £ ¶ '4 òÑ
êÚål ²²þÂñ nõª Bessel Ø ª AíØ íØ 2 f(x) 3 [−π, π] þëY, f(x) n¼êX 1, cos x, sin x, · · · , cos nx, sin nx, · · · ¥z¼êÑ, K f(x) ≡ 0. y² d^ f(x) Fourier XêÑ 0, Ï d Parseval ª Z π −π f 2 (x) dx = 0. du f(x) ´ëY¼ê, þª%¹ f(x) ≡ 0. íØ 3 f(x) Ú g(x) Ñ3 [−π, π] þëY, ¦kÓ Fourier Xê, K f(x) ≡ g(x). 10/17 kJ Ik J I £ ¶ '4 òÑ