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2 2 2 L 5 2. L L L L k I 5 4. I I [](E.M.,R ( ) ) L 2 ( ) 2

3 ˆf(ξ) f(x)e 2πiξx dx (ξ R) f f(x) dx < L L L { ˆf : f L (R)} ([2, p. 69] ) L ( 2., 7). R ϕ(z) z < R ϕ(0) 0 L h L (R) h < R ϕ(ĥ) ĝ g L (R) ϕ(ĥ) L L( ) L ( 2.2). f(x) + x 2 k f k (x) (f f f)(x) kπk x 2 + k 2. f(x) ˆf(ξ)e 2πiξx dx (x R) ˆf(n) f(n) n Z n Z L 2 2

4 ζ(s) n n s (s C) ζ(2), ζ(4),.... ζ(2) π2 π4, ζ(4) ζ() (n + ) (n + 2) 4 24 π (4n + ) (4n + ) π 2 L I 2 L L 4 I 5 6 4

5 2 L 2. L [4, ] [5] (L ). R f f f(x) dx < L (R) L f L (R) f L f L L 2 ( ). f L (R) ˆf(ξ) ˆf f L L f(x)e 2πiξx dx (ξ R) L { ˆf : f L (R)} F L (R) L L 2 ( ). [, ] f(x) { ( x ) 0 ( < x ) f(x) L ˆf(ξ) f(x)e 2πiξx dx e 2πiξx dx sin 2πξ πξ f(x) ˆf(ξ) ξ 0 ( ) 5

6 ˆf L (R) 2 ( ). L f L (R) ˆf L (R) f(x) ˆf(ξ)e 2πiξx dx (x R) [ ]. [2, p. 44, ] ( 9). F [ ]. α, β C f, g L (R) F(αf + βg) αf(f) + βf(g) () F(αf + βg)(ξ) () α (αf(x) + βg(x))e 2πiξx dx αf(x)e 2πiξx dx + f(x)e 2πiξx dx + β αf(f)(ξ) + βf(g)(ξ). f R ˆf(ξ) [ ]. [2, p. 2,(.5)] βg(x)e 2πiξx dx g(x)e 2πiξx dx 2. L {f n } n L (R) {f n } n f L ˆ f n ˆf R 6

7 [ ]. f f n 0 (n ) f ˆ n (ξ) ˆf(ξ) (f n f)ˆ(ξ) (f n f)(x)e 2πiξx dx f n f (f n f)(x) dx sup ξ R ˆ f n (ξ) ˆf(ξ) f n f 0 (n ) ˆ f n ˆf R 2.2 L L L F(L (R)) F L L 4 ( ). f L (R) ˆf(ξ) 0 ( ξ ) [ ]. [2, Theorem]. [, ] ˆf(ξ) sin 2πξ πξ sin 2πξ lim ξ πξ 0 5. L f(x) ˆf(ξ) < b < b ˆf(ξ) ξ dξ A f b 7

8 [ ]. ˆf(ξ) ˆf(ξ) ˆf( ξ) b 2πx f(x) cos(2πξx)dx 0 ˆf(ξ) i f(x) sin(2πξx)dx ˆf(ξ) dξ ξ b { } { b f(x) sin(2πξx)dx dξ ξ f(x) 2πxb sin(u) f(x) u du dx (2πxξ u ) A f(x) dx A f } sin(2πξx) dξ ξ dx 4. 0 L g(x) / log(x) (e < x) g(x) x/e ( x < e) / log( x) (x < e) g(x) b g(x) dx x b g(ξ) ξ dξ e b e ξ ξ dξ + b e ξ log(ξ) dξ + log(log(b)) (b ) e g(ξ) a ξ dξ 5 g L C 0 (R) {g(ξ) : g R g(ξ) 0 ( ξ )} L C 0 (R) L C 0 (R) 8

9 2. ( ). f, g L (R) f g (f g)(x) f(x t)g(t)dt (2) f g f g f f f k f k f (k 2) f k f k. f, g L (R) [ ]. () f g f g f g () f g g f (4) (f g)(x) dx f(x t)g(t)dt dx (5) x t s (5) ( f g f(x t)g(t) dtdx (5) f(s)g(t) dtds ) ( ) f(s) ds g(t) dt 9

10 (5) (4) (f g)(x) (4) (g f)(x) f(x t)g(t)dt f(u)g(x u)( du) (x t u ) f(u)g(x u)du 6. f, g L (R) (f g)ˆ(ξ) ˆf(ξ) ĝ(ξ) [ ]. (f g)ˆ(ξ) (f g)(x)e 2πiξx dx { { g(t) } f(x t)g(t)e 2πiξx dt dx (6) x t p { (6) g(t) ( ) ( g(t)e 2πiξt dt ˆf(ξ)ĝ(ξ) } f(x t)e 2πiξx dx dt (6) } f(p)e 2πiξ(t+p) dp dt ) f(p)e 2πiξp dp k 0

11 . f L (R) ˆf k (ξ) ( ˆf(ξ)) k 6 L (L (R), +, ) (L, +, ). L L L L 7. R ϕ(z) z < R ϕ(0) 0 L h L (R) h < R ϕ(ĥ) ĝ g L (R) ϕ(ĥ) L [ ]. ϕ(z) z < R R ϕ(z) a k z k (7) k (ϕ(0) 0 a 0 0) h < R ξ ĥ(ξ) h(x)e 2πiξx dx h(x) dx h < R ĥ(ξ) < R (7) z ĥ(ξ) ϕ(ĥ(ξ)) a k (ĥ(ξ))k a k (ĥk(ξ)) k k (ξ R). a k h k L n m k

12 (n, m Z) n m n a k h k a k h k a k h k k k km h < R n, m n a k h k km n a k h k km n a k h k km n a k h k 0 (n, m ) km L (R) L n a k h k g 0 (n ) g L (R) 2 ĝ(ξ) a k ĥ k (ξ) k a k (ĥ(ξ))k ϕ(ĥ(ξ)) k k.2 L k 7 5. f(x) + x 2 k f k (x) kπk x 2 + k 2 f 2 (x), f (x), f 4 (x),... f 2 (x) dt f(x t)f(t)dt ( + (x t) 2 )( + t 2 ) { 2 x(x 2 +4) t + x (x t) x(x 2 +4) t + } x t 2 dt 2π x

13 f (x), f 4 (x),... 7 f(x) + x 2 e 2πizξ ξ > 0 Γ dz + z2 Γ Γ e 2πizξ Γ z i + z2 ( ) e 2πizξ Res + z 2 : z i e 2πξ 2i e 2πizξ ( ) e 2πizξ dz 2πi Res + z2 + z 2 : z i πe 2πξ Γ Γ [ R, R] γ R Γ e 2πizξ R + z 2 dz e 2πixξ R + x γr 2 dx + e 2πizξ + z 2 dz γ R R 0 + R 2 e 2iθ R 2 { sin θ 2 π θ + 2 (π θ 2 π) sin θ 2 π θ 4 ( 2 π < θ 2π)

14 z Re iθ (θ : 2π π) + z 2 dz 2π R π { 2 π e 4Rξθ+4πξR dθ + γr e 2πiξz R R 2 R R 2 π {e 4πξR [ 4ξR e 4ξRθ 2πξR sin θ e + R 2 e 2iθ dθ ] 2 π π 2π e 4Rξθ 8πξR dθ 2 π } + e 8πξR [ 4ξR e4ξrθ R { e 4πξR 4ξR(R 2 ( e 6πξR + e 4πξR ) + e 8πξR (e 8πξR e 6πξR ) } ) 2ξ(R 2 ) ( e 2πξR ) 0 (R ) e 2πixξ ξ > 0 + x 2 dx πe 2πξ ξ < 0 R ˆf(ξ) πe 2πξ ξ 0 dx π + x2 ξ R ˆf(ξ) πe 2π ξ ϕ(z) e z ϕ(z) (8) t ˆf(ξ) ϕ(z) k z k k! ] 2π 2 π ( z < ) (8) } ϕ(t ˆf(ξ)) (t ˆf(ξ)) k k! k k ˆf k (ξ) t k k! e tπe 2π ξ k ˆf k (ξ) t k (9) k! (9) t (πe 2π ξ )e tπe 2π ξ k2 ˆf k (ξ) (k )! tk (0) (0) t 0 ˆf 2 (ξ) πe 2π ξ 4

15 (0) k t 0 ˆf k (ξ) π k e 2πk ξ f k (x) kπk x 2 + k 2 4 I 4. I 4 ( I). a > 0 f I a. f S a {z C : Im(z) < a} 2. K > 0 x R y < a f(x + iy) K + x 2 I a>0 I a 8. a > 0 f I a f R L 0 b < a ˆf L ˆf(ξ) Be 2πb ξ [ ]. b 0 ˆf(ξ) f(x)e 2πiξx dx f(x) dx K dx Kπ + x2 0 < b < a ξ > 0 g(z) f(z)e 2πiξz 2 5

16 2 (i),(ii),(iii),(iv) R g (ii) (iv) 0 (iv) z R it (t : b 0) f(x + iy) (iv) b 0 g(z)dz 0 b b g( R it)( idt) g( R it) dt f( R it)e 2πiξ( R it) dt b K K f( R it) + x2 + R 2 K R 2 () b 0 K R 2 e 2πξt dt 0 0 f( R it)e 2πξt dt () K 2πξR 2 ( e 2πξb ) 0 (R ) (iv) 0 (ii) (i) R R f(x)e 2πiξx dx R ˆf(ξ) (iii) z x ib (x : R R) R R g(z)dz g(x ib)dx f(x ib)e 2πi(x ib)ξ dx (iii) R R g(z)dz 0 R ˆf(ξ) (i)+(ii)+(iii)+(iv) f(x ib)e 2πi(x ib)ξ dx 6

17 ˆf(ξ) f(x ib)e 2πi(x ib)ξ dx K + x 2 e 2πbξ dx Be 2πbξ ξ < 0 ξ < I 9 ( ). f I x R f(x) ˆf(ξ)e 2πiξx dx 2 [ ]. ξ ˆf(ξ)e 2πiξx dx 0 ˆf(ξ)e 2πiξx dx + 0 ˆf(ξ)e 2πiξx dx 2 f I a 0 < b < a 8 ˆf(ξ) f(u ib)e 2πi(u ib)ξ du 7

18 ib L {u ib : u R} 0 ˆf(ξ)e 2πiξx dξ 0 2πi 2πi { } f(u ib)e 2πi(u ib)ξ du e 2πiξx dξ { } f(u ib) e 2πi(u ib x)ξ dξ du 0 f(u ib) 2πb + 2πi(u x) du L f(u ib) u ib x du f(ξ) dξ (2) ξ x ξ < 0 ib L 2 {u + ib : u R} 0 ˆf(ξ)e 2πiξx dξ 2πi L 2 x R γ R f(ξ) dξ () ξ x 4 γ R f(x) 2πi γ R f(ξ) ξ x dξ 8

19 R 0 f(x) f(ξ) 2πi ξ x dξ f(ξ) 2πi ξ x dξ 0 L ˆf(ξ)e 2πiξx dξ + ˆf(ξ)e 2πixξ dξ 0 L 2 ˆf(ξ)e 2πiξx dξ ( (2), () ) 0 ( ). f I n Z f(n) n Z ˆf(n) [ ]. f I a 0 < b < a 2πi f(z) e 2πiz e 2πiz z n Z n f(n) 2πi N Z γ N 5 γ N 2πi n N f(n) 2πi γ N f(z) e 2πiz dz 9

20 w > f(n) n Z L f(z) e 2πiz dz L 2 w w L z x ib w < w n e 2πiz e 2πi(x ib) e 2πb > e 2πiz e 2πiz w w n e 2πinz L 2 e 2πiz < e 2πiz e 2πinz f(z) e 2πiz dz (4) (4) ( ) ( ) f(n) f(z) e 2πiz e 2πinz dz + f(z) e 2πinz dz n Z L L 2 f(z)e 2πi(n+)z dz + f(z)e 2πinz dz L L 2 f(x)e 2πi(n+)x dx + f(x)e 2πinx dx ˆf(n + ) + ˆf(n) n Z ˆf(n) 5 20

21 5. 5 ( ). s C (Re(s) > ) ζ(s) ([, p. 72, 2.4] ) ζ(2) ζ(4). τ C Z n Z n n Z n s (τ + n) 2 π 2 sin 2 (πτ) (5) (τ + n) π cos(πτ) sin (πτ) (6) (τ + n) 4 π4 (cos(2πτ) + 2) sin 4 (πτ). τ C (Im(τ) > 0) k 2 n [ ]. ξ > 0 f(x) (τ + n) k ( 2πi)k (k )! (7) m k e 2πimτ (8) m (τ + x) k ˆf(ξ) e 2πiξx (τ + x) k dx 2

22 Γ Γ e 2πiξz (τ + z) k dz 6 Γ e 2πiξz z τ k (τ + z) k ( ) e 2πiξz Res (τ + z) k : z τ lim z τ Γ Γ (k )! ( 2πiξ)k e 2πiξτ (k )! e 2πiξz ( 2πi)k dz (τ + z) k (k )! Γ d k ( ) e 2πiξz dz k (τ + z)k (τ + z) k ξ k e 2πiξτ e 2πiξz (τ + z) k dz γ R e 2πiξz R (τ + z) k dz R e 2πiξx (τ + x) γr k dz + e 2πiξz (τ + z) k dz γ R R 0 τ + Re iθ (R τ ) k { sin θ 2 π θ + 2 (π θ 2 π) sin θ 2 π θ 4 ( 2 π < θ 2π) 22

23 z Re iθ (θ : 2π π) (τ + z) k dz 2π 2πξR sin θ R e π τ + Re iθ k dθ { 2 π e 4Rξθ+4πξR dθ + γr e 2πiξz R (R τ ) k R (R τ ) k π {e 4πξR [ 4ξR e 4ξRθ ] 2 π π 2π e 4Rξθ 8πξR dθ 2 π } + e 8πξR [ 4ξR e4ξrθ R { e 4πξR 4ξR(R τ ) k ( e 6πξR + e 4πξR ) + e 8πξR (e 8πξR e 6πξR ) } 2ξ(R τ ) k ( e 2πξR ) 0 (R ) ξ > 0 ξ < 0 ˆf(ξ) ( 2πi)k ξ k e 2πiξτ (k )! Γ ] 2π 2 π e 2πiξz (τ + z) k dz Γ } 7 Γ Γ e 2πiξz (τ + z) k Γ e 2πiξz (τ + z) k dz 0 2

24 ξ > 0 γ R Γ e 2πiξz R (τ + z) k dz R e 2πiξx (τ + x) k dz + γ R e 2πiξz (τ + z) k dz γ R γ R γ R e 2πiξz dz 0 (R ) (τ + z) k ξ < 0 e 2πiξx (τ + x) k dx 0 ξ 0 f(x) [ dx (τ + x) k ] (τ + x) (k ) 0 (k ) (τ + x) k ( 2πi) k ˆf(ξ) (k )! ξk e 2πiξτ (ξ > 0) 0 (0 ξ) n. z < (τ + n) k ( 2πi)k (k )! z n n z z nz n n m k e 2πimτ m z z z ( z) 2 (9) 24

25 (9) z z n n 2 z n n (8) k 2 n z(z + ) ( z) (20) n z n z(z2 + 4z + ) ( z) 4 (2) (τ + n) 2 4π2 (9) n (τ + n) 2 4π2 k k 4 n (τ + n) ( 2πi) 2! m(e 2πiτ ) m m m(e 2πiτ ) m m 4π 2 e 2πiτ ( e 2πiτ ) 2 4π 2 (e πiτ e πiτ ) 2 π 2 sin 2 (πτ) 8π i 2 π 2 m e 2πiτ (e 2πiτ + ) ( e 2πiτ ) m 2 (e 2πiτ ) m 4π e πiτ + e πiτ i (e πiτ e πiτ ) π cos(πτ) sin (πτ) n (τ + n) 4 ( 2πi)4! 8π4 8π4 π 2 m m (e 2πiτ ) m e 2πiτ ((e 2πiτ ) 2 + 4e 2πiτ + ) ( e 2πiτ ) 4 e 2πiτ e 2πiτ (e πiτ e πiτ ) 4 π4 (cos 2πτ + 2) sin 4 πτ 25

26 Im(τ) > 0 τ C Z (5) n (τ + n) 2 π 2 sin 2 (πτ) τ n Z 2 C Z (5) τ C Z (6) (7) ζ(2) ζ(4) (5) n Z {0} (τ + n) 2 π2 sin πτ 2 τ 2 π2 (τ 0) n Z {0} (τ + n) k 2 (τ + n) k (k : ) (22) ζ(2) n n n 2 π2 6 ζ(4) n Z {0} (τ + n) 4 π4 (cos(2πτ) + 2) sin 4 (πτ) τ 4 π4 45 (τ 0) ζ(4) ζ(4) n n 4 π ζ(2) ζ(4) (22) (6) 26

27 2. (n + ) (n + 2) 4 24 π 4 5 π (2) (4n + ) (4n + ) π 2 π 2 5 (24) [ ]. (2) (6) τ n Z ( + π cos( π ) n) sin ( π ) n (n + ) 4 24 π 4 5 π n Z n (n + ) (n + 2) (n + ) (n + 2) 4 24 π 4 (24) (6) τ 4 n n n Z (4n + ) π 2 π 2 5 (4n + ) (4n + ) 5 π (4n + ) (4n + ) π 2 π

28 6. f(x) + x 2 k f k (x) kπk x 2 + k n Z n (τ + n) 2 π 2 sin 2 (πτ) (τ + n) 4 π4 (cos(2πτ) + 2) sin 4 (πτ) ζ(2) π2 π4, ζ(4) 6 90 ζ(6), ζ(8),.... (n + ) (n + 2) 4 24 π 4 (4n + ) (4n + ) π 2 π π n Z (τ + n) π cos(πτ) sin (πτ) τ, 4 ζ() ζ() 28

29 L L L I f I L [] E.M. R 2009 [2] K,Chandresekharan Classical Fourier Transforms Springer-Verlag 987 [] Elias M. Stein, Guido Weiss Introduction to FORIER ANALYSIS ON EU- CLIDEAN SPACES Princeton University Press 97 [4] 2000 [5] 96 29

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