Riemann-Lebesgue ùÜDÿY¹
Šœþ
‡k:
BbÊ¥¹dıø* Riemann - Lebesgue ùÜ}–, ¥Ê¡H}&2rÆ7
”½b5iH 7(}D…óÉí3æ¨
ìó¶(method of stationary phase) DÿY¹Ô u Young’s¿ 5–1
Riemann - Lebesgue ùÜ5–Ä@ç u Fourier b5Y¹4½æD Fourier }t, ²Æuzÿuƒ Fourier |‚
à}׉b¶jÏfûj˙, |(Ûbû˝
íL½æ (inverse problem), Ĥªcw½ b4 Bbø# _òh<2í„p, 7(
} semi - classical í–1, |(†ùªÿY
¹D Young’s ¿ Ê¡H}&«wuÝ(
4R}j˙2”½bí–1
1. Riemann - Lebesgue ùÜ
} Riemann-Lebesgue ùÜJ.*U
˚u“bç2í˹”—Fourier bOGÿ .øwŸá<2 #ìøƒb f ∈ C[−π, π]
ø5[Ñ Fourier b f (x) ∼ a 0
2 +
∞ n =1
(a n cos nπ
L x+b n sin nπ L x) (1.1)
a n = L 1 −L L f (ξ) cos nπ L ξdξ
b n = L 1 −L L f (ξ) sin nπ L ξdξ (1.2)
‚àúiƒbí¸Ï“†,ª[Ñ
f (x) = 1 2L
L
−L f (ξ)dξ +
∞ n =1
1 L
L
−L f (ξ) cos nπ
L (x −ξ)dξ (1.3) BbÛÊ>Eí½æu: ¦ x ì7I L → ∞ † (1.3) }AÑBóá? çÍ Ñ7\„}íæÊ4, Bbílcq f ∈ L 1 ( R ) ¹
∞
−∞ | f(ξ) | dξ < ∞ (1.4) Ĥ (1.3) AÑ (¤v 2L 1 −L L f (ξ)dξ → 0)
f (x) = lim
L →∞
∞ n =1
1 L
· L
−L f (ξ) cos nπ
L (x − ξ)dξ (1.5) ÛÊ_" Riemann ¸íj¶I
λ n = nπ
L ∆λ = λ n +1 − λ n = π L I(λ, L) =
L
−L f (ξ) cos λ(x −ξ)dξ (1.6) Ĥ (1.5) ªZŸÑ
f (x) = lim
L →∞
1 π
∞ n =1
I(λ n , L)∆λ (1.7)
17
¥£u Riemann ¸5$ I∆λ → 0 (L → ∞) †
f (x) = lim
L →∞
1 π
∞
0 I(λ, L)dλ
= lim
L →∞
1 π
∞
0 dλ
L
−L f (ξ) cos λ(x −ξ)dξ
= 1 π
∞
0 dλ
∞
−∞ f (ξ) cos λ(x −ξ)dξ (1.8)
¥_ÿu|±íFourier}t y‚à¸
Ï“ø (1.8) ZŸÑ f (x) = 1
π
∞
0 dλ
∞
−∞ f (ξ) cos λ(x −ξ)dξ
= 1 π
∞
0 [a(λ) cos λx+b(λ) sin λx]dλ (1.9) w2
a(λ) = −∞ ∞ f (ξ) cos λξdξ
b(λ) = −∞ ∞ f (ξ) sin λξdξ (1.10)
¥£u Fourier ìý (£ý) ‰² wõÿu Fourier b5R2 çÍBbª‚à Euler tøs6¯9Êø–
f (x) = 1 π
∞
0 dλ
∞
−∞ f (ξ) cos λ(x −ξ)dξ
= 1 2π
∞
0 dλ
∞
−∞ f (ξ)e iλ (x−ξ) dξ + 1
2π
∞
0 dλ
∞
−∞ f (ξ)e −iλ(x−ξ) dξ
= 1 2π
∞
−∞ dλ
∞
−∞ f (ξ)e iλ (x−ξ) dξ
=
∞
−∞
f (λ)e ˆ iλx dλ (1.11) w2
f (λ) = ˆ 1 2π
∞
−∞ f (ξ)e −iλξ dξ (1.12)
ÿuƒb f 5 Fourier ‰² Ĥ Fourier
}t5…”ÿuø/L‰² —Fourier L‰²7¥£ujR}j˙, Ô u(4 4[bR}j˙|½bí5x
,Þ¥<Rû¬˙2_u°íÛïÿ u·|Û6-é5}
b
a
f (x)
cos nx sin nx
dx,
b
a
f (x)e iλx dx a.b ∈ R
ĤÃ7í„p5‡âln6¥é}í 4”, ¥ÿu Riemenn - Lebesgue ùÜ:
Riemann - Lebesgue ùÜ:
I:(ä–È)
J f ∈ L 1 ([0, 2π]) †
n →±∞ lim 1 2π
2π
0 f (x) cos nxdx
= lim
n →±∞
1 2π
2π
0 f (x) sin nxdx = 0 C[ѵb5$
n →±∞ lim f (n) ˆ
= lim
n →±∞
1 2π
2π
0 f (x)e −inx dx = 0 (1.13)
II:(Ìä–È)Jf ∈ L 1 ( R ) †
λ →±∞ lim f (λ) ˆ
= lim
λ →±∞
1 2π
∞
−∞ f (x)e −iλx dx = 0 (1.14) â}5©/4Bbyª!6
f ∈ L 1 ( R ) ⇒ ˆ f ∈ C 0 ( R ) (1.15)
C 0 ( R ): [ýF©/ƒbÅ—Ê̤õÑ 0
5Õ¯
Ék Riemann - Lebesgue ùÜ|o uâ Riemann k 1876 -„p, 7øOí8
$¹ f ∈ L 1 †uâ Lebesgue k 1903 - F#í Ék¥ìÜí„prÖ/, OBb 1Ì<J¥/jú&¥1(/½b5ìÜ Äуb f 5¸ˇØ2 (Bý©/) FJb }&í.u¥á7u cos nx, ÑBóá? B bõõwÇ$, * Síi Vpëw<2
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−1 1
2π
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cos x (1 _ š) Ç ø
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−1 1
2π
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cos 3x (3 _ š) Ç ù
âÇ$ªø cos nx Ñ n _ìýšÑÊ [0, 2π] ¥_–È, Ĥç n → ∞ v ª;díu cos nx Ê 1 D −1 5 È ( ÄÑ | cos nx| ≤ 1) 0§PÓ
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cos nx(n _ š) 2π
· · ·
7Ékw} 0 2π cos xdx = 0, †Ä Ñ cos x ÑøU‚ 2π íƒb, w}ø
¶MÊ x W5,j, Çø¶M†Ê x W
5-j ¥s¶Mø£øŠ óJ¾, Ĥ
2π
0 cos xdx = 0 ° Üúƒb cos nxø£
øŠÉuÛÊ)òø<, ˛¤ óJ¾, ]
2π
0 cos nxdx = 0 “0” ¥_Mu<2í,
…ÿu cos x íMÌM 1
2π
2π
0 cos xdx = 0
*Ç$,Võ cos nx ¥<ƒbJ x WÑ2 -7,-PÓ, ]w}¹MÌMÑ 0, °Ü úk}
2π
0 f (x) cos nxdx
5n6D‡Þêró°, ñøÏ íuPÙ Z‰Ñ |f(x)| (Ÿlu1)
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2π f (x)
−f(x) Ç ú
¤v} 0 2π f (x) cos nxdx . c)bÑ 0 Oç n → ∞ v£íDŠí¶M ˛˛¤J
¾, Ĥ¯Üí“¿u
n lim →∞
2π
0 f (x) cos nxdx = 0 à‹ cos x H²Ñ sin x, 6ó°í!‹
n lim →∞
2π
0 f (x) sin nxdx = 0
¥ÿu Riemann - Lebesgue ùÜ
ìÜ„p:
M Bbø} J|1zpwÊbç2&, í½b4
(1) Riemann ¸:
ʇÞín62Bb˛%ø−Ê–È [0, 2π], cos nx â n _ìýšF A, Ĥ
Bbø [0, 2π] }A n } (¥_–1ÿu Riemann ¸)
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2π
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2π n
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4π n
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2(n−1)π n
Ç û
2π
0 → n
k =1
2k nπ
2(k−1)
n
π
(1.16) Ĥ}ª[Ñ
2π
0 f (x) cos nxdx
=
n k =1
2k nπ
2(k−1)
n
π
f (x) cos nxdx
=
n k =1
2kπ
2(k−1)π f ( y
n )(cos y) 1 n dy (x = y
n )
= 1 n
n k =1
2π
0 f (ξ + 2(k − 1)π
n ) cos ξdξ (y = ξ + 2(k − 1)π)
= 1 2π
2π
0 cos ξ
·( n
k =1
f ( ξ + 2(k − 1)π
n ). 2π
n )dξ O f Ñ©/ƒb, w Riemann Å—
n k =1
f ( ξ + 2(k − 1)π
n ) 2π
n
→ 2π
0 f (x)dx, n → ∞ ]ø
2π
0 f (x) cos nxdx
→ ( 1 2π
2π
0 cos ξdξ)(
2π
0 f (x)dx)
= 0 (1.17)
Ê„pí¬˙2ªJüìíu}í”
ÌM kÉ1Ý.Íí (44Bb}¥ó“), 7uÄ Ñ cos ξ 5MÌM (mean) ÑÉíí ] à‹J cos 2 x H cos x ª)
2π
0 f (x) cos 2 nxdx
→ ( 1 2π
2π
0 cos 2 ξdξ)(
2π
0 f (x)dx)
= 1 2
2π
0 f (x)dx (1.18)
çÍ6ª‚àìÜí!‹V„p
2π
0 f (x) cos 2 nxdx
=
2π
0 f (x) 1 + cos 2nx 2 dx
= 1 2
2π
0 f (x)dx+ 1 2
2π
0 f (x) cos 2nxdx
1 2
2π
0 f (x)dx + 0 = 1 2
2π
0 f (x)dx BbêÛøKí9õ:
( lim
n →∞
2π
0 f (x) cos nxdx) 2
= lim n →∞ 2π
0 f (x) cos 2 nxdx (1.19)
BbÊ|(øJy‚àÿY¹ (weak con-
vergence) íh1V}¤Ûï
Ê,Hí„p¬˙2, Bbcàƒ cos x ÑøU‚Ñ 2π íƒb, ĤªòQR2BL
<U‚°Ñ 2π í©/ƒb g(x) :
n lim →∞
2π
0 f (x)g(nx)dx
= 1 2π
2π
0 g(x)dx 2π
0 f (x)dx (1.20)
¤v cos x â g(x) ¦H; cos nx †â g(nx) ≡ g n (x) ¦H
2π
0 f (x)g n (x)dx
=
2π
0 f (x)g(nx)dx
=
n k =1
2kn
π
2(k−1)
n
