John K. Hunter nd B. Nchtergele, Applied Anlysis, World Scientific, 21. Wine cellr Applied Anlysis M. Reed nd B. Simon, Functionl Anlysis, Acdemic Pre

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2 John K. Hunter nd B. Nchtergele, Applied Anlysis, World Scientific, 21. Wine cellr Applied Anlysis M. Reed nd B. Simon, Functionl Anlysis, Acdemic Press, 198. (fourier22, integrl27) (hilbert212) ymgmi/teching/functionl/hilbert212.pdf ( ) 2

3 A 49 B 5 C Plncherel formul 52 D 53 3

4 1 e iθ = cos θ + i sin θ, cos θ = eiθ + e iθ, sin θ = eiθ e iθ 2 2i f(t) = ce iωt 1. sin 3 θ sin θ, sin(3θ) d 2 f dt 2 + ω2 f =, (periodic function (period) T f(t + T ) = f(t). ω = 2π/T (frequency) 1/T e iωt T = 2π/ω 2. e iωt T > ω [, T + ] x ( π < x < π) 2π x ( π < x < π) θ = 2πt/T T = R/T Z, e iθ t + T Z 3. R d / j T jz = j R/T jz (periodicl integrtion) T f(t) dt = +T f(t) dt. t f(t) f(t) = g(t) + ih(t) b b f(t) dt = b f(t) dt = lim g(t) dt + i n j=1 b h(t) dt n f(τ j )(t j t j 1 ) b f(t) dt = F (b) F (), F (t) = f(t) 4

5 b f(t) dt b f(t) dt ( b) (e iωt ) = iωe iωt n Z 2π e int dt = { 2π if n =, otherwise 4. * e (+ib)t e t cos(bt) dt, e t sin(bt) dt 2 T f(t) 2 dt < + (squre integrble) T H T H T [, T + ] L 2 (, T + ) f(t) + g(t) 2 2( f(t) 2 + g(t) 2 ) f, g H T = αf + βg H T H T 2 f(t)g(t) f(t) 2 + g(t) 2 (f g) f(t)g(t) dt = T T + f(t)g(t) dt 5. z, w z + w 2 2( z 2 + w 2 ), 2 zw z 2 + w 2 5

6 (i) (f g 1 + g 2 ) = (f g 1 ) + (f g 2 ), (f βg) = β(f g). (ii) (f 1 + f 2 g) = (f 1 g) + (f g 2 ), (αf g) = α(f g). (iii) (f g) = (g f). (iv) (f f). 6. (f f) = = f = (inner product) (non-degenercy) *1 (Schwrz inequlity) (f g) 2 (f f) (g g) T f(t)g(t) dt 2 f(t) 2 dt g(t) 2 dt T T (f f) = f *2 H T (f g) 7. * f (f f) = f(t) = ( t) f b 8. [, b] f b b f(t) dt 1 dt f(t) 2 dt b f(t) dt < +, b f(t) 2 dt = + (inner product spce) v = (v v) (v w) v w v + w v + w d(v, w) = v w (Hilbert spce *3 ) *1 Hermnn Schwrz Cuchy Bunykovski *2 T f(t) g(t) 2 dt = f, g *3 Dvid Hilbert ( ) 6

7 9. 1. e t ( t π, T = 2π) (i) (ii) (X, µ) L 2 (X, µ). l 2 (Z) (X = Z, µ = counting mesure) Ω R n L 2 (Ω) (X = Ω, µ = Lebesgue mesure). (iii) H T. 11. l 2 (N) {e i } i I (e i e j ) = δ i,j, i, j I (orthonorml system) 2.2. {e int / 2π} n Z H 2π = L 2 (, 2π) {cos(nt)/ π, sin(nt)/ π} n=1,2,... 1/ 2π H 2π = L 2 (, 2π) 12. n π sin 2n (x) dx 13. T 2.3 (). V {e k } 1 k n v V n v = v (e k v)e k (v e k ) = (k = 1, 2,..., n) {z k } 1 k n n n v z k e k 2 = v 2 + z k (e k v) 2 k=1 k=1 k= V {e i } i I v V (e i v) 2 (v v) = v 2. i I 7

8 Bessel (Bessel s inequlity) 14. * x, x 2, x 3, ( x π) Grm-Schmidt f 1, f 2, f 3 sin x ( π x π) π π, b, c sin x x bx 2 cx 3 2 dx 2.5 ( ). [, b] f(t) b lim n ± f(t)e int dt =. Proof. [, b] [ π, π] f t [ π, π] \ [, b] b f(t)e int dt = 2π(e n f) (n ± ) [, b] [ π, π] [, b] [ π + 2πk, π + 2πk] (k Z) π+2πk π+2πk f(t)e int dt = = π π π 15. f(t) = 1, f(t) = t π f(s + 2πk)e in(s+2πk) ds f(s + 2πk)e ins ds Remrk. lim f(t) cos(nt) dt =, n f(t) sin(nt) dt = lim n cos(nt) sin(nt) f (mplitude modultion) f f cos(nt), sin(nt) 2π/n 3 2π f(x) f(x) 2 dx < + 2π f n e inx, f n C n Z 8

9 (Fourier series) f(x) 2π f n = 1 f(x)e inx dx, n =, ±1, ±2,... 2π 2π f n f (Fourier coefficient) 2π 2π 2π f(t)e int dt 1dt f(t) 2 dt < {e n (x) = e inx / 2π} (e n f)e n (x) n Z f(x) f(x) = lim n k= n n (e k f)e k (x), x R n k= n 1 2π 2π f(y)e ik(x y) dy = 1 2π 2π D n (x y)f(y) dy, D n (x) = sin((2n + 1)x/2) sin(x/2) D n n 16. n D n (x) = k= n e ikx n D n 3.1. f(x) = f = 1 2, { 1 if x < π, if π x <. f n = 1 ( 1)n 2πin f n e inx n Z 9

10 17. f(x), g(x) (f g) f k, g k 18. cos(mx), sin(mx) 19. f 2. m x m + = { x m if x, otherwise [ π, π] Remrk. (regulriztion) Fejer Poisson lim f n = n ± < r < 1 f n r n e inx n Z r 1 f n f 1 π f(y)p r (x y) dy 2π π P r (x) P r (x) = r n e inx = (re ix ) n + (re ix ) n n Z = n= n=1 1 re ix + 1 reix 1 re ix = 1 r 2 1 2r cos x + r 2 2π Poisson (Poisson kernel) Poisson P r 1 r 2 P r (x) = (1 r) 2 + 4r sin 2 x (Poisson ). (i) P r (x) x 1+r 1 r P r(x) 1 r 1+r 1

11 (ii) (iii) 1 π P r (x) dx = 1, 2π π lim P r(x) = r 1 for x. More precisely, δ >, ϵ >, r < 1, P r (x) ϵ for δ x π nd r r < * P r (x) π f(x) {f n } f(x) = x lim r 1 n Z f n r n e inx lim r 1 sup{ f(x) n f n r n e inx ; x R} =. Proof. ϵ > f(x) f(y) ϵ for x y δ δ > P r (x y) ϵ if x y δ r < 1 1 π 2πf(x) π f(y)p r (x y) dy = (f(x) f(y))p r (x y) dy π = ϵ π x y δ π π π π f(x) f(y) P r (x y) dy f(x) f(y) P r (x y) dy + 2πϵ + 4Mπϵ P r (x y) dy + ϵ π π x y δ f(x) f(y) dy M = f = sup{ f(x) ; x R} f(x) f(y) P r (x y) dy 3.4 (). ϵ >, N, { n } N n= N N f n e n = sup f(x) N x R N n= N n 2π e inx ϵ. h(x) (x R) h = sup{ h(x) ; x R} 11

12 3.5. 2π f(x) H 2π = L 2 (, 2π) f = n Z(e n f)e n *4 (summble) ϵ >, F Z F F Z f (e n f)e n ϵ n F lim M,N f N n= M (e n f)e n = Proof. f f N n= N 2 (e n f)e n f N n= N 2 n e n 2π f n e n 2 lim N f N n= N (e n f)e n 2 = f g f g N f (e n f)e n f g (e n f g)e n + g (e n g)e n n= N f g + g (e n g)e n * f g f g π f(x), g(x) (f g) = (f e n )(e n g) π π f(x)g(x)dx = 2π n Z f n g n, f n = 1 π e inx f(x) dx. 2π π *4 12

13 π π f(x) 2 dx = 2π f n 2, (f f) = (e n f) 2 n Z n Z Proof. Remrk. (convergenc in men), pointwise convergence H {e n } v (v v) = n (e n v) 2 (complete) Prsevl (Prsevl s equlity) f = n (e n f)e n (orthonorml bsis) (f g) = n (f e n )(e n g) I = n e n )(e n (Dirc ) 24. T > 25. y R x k e iyx dx e iyx dx = i y e iyx y xe iyx dx = ix y e iyx + 1 y 2 e iyx x 2 e iyx dx = i x2 y e iyx + 2x y 2 e iyx 2i y 3 e iyx 13

14 x, x 2 ( π < x < π) i n ( 1)n (n ), 2 n 2 ( 1)n (n ) Prsevl ζ(2) = ζ(4) = n=1 n=1 1 n 2 = π2 6 1 n 4 = π x 2 x 3 4 F (x + 2π) = F (x) F f(x) = [,x] F (t) dt + C [, ) F f(x + 2πn) = n F (t) dt + f(x), x, n =, 1, 2,... f 2π F (t) dt =. 2π f(x) F F f F = f 4.1. g(x) = x ( x π) G(x) = x/ x ( π < x π) F (x) = 1 ( x π) f(x) = x + C π π G(x) dx =, π π F (x) dx = 2π. Remrk. x ( π < x π) h(x) δ(x) H(x) = 1 2π n Z δ(x 2πn π) h h 4.2. f F [, b] C 1 g(x) b g (x)f(x) dx = b g(x)f (x) dx + f(b)g(b) f()g(). 14

15 Proof. b g (x)f(x) dx = = = b b b = ( x g (x) ( b b F (y) y ) F (y) dy + C ) g (x) dx dx dy + C b g (x) dx F (y)(g(b) g(y)) dy + C(g(b) g()) F (y)g(y) dy + f(b)g(b) f()g() f F f C 1 F f Proof. F f E f b (E(x) F (x))g(x) dx = C 1 g g(x) g(x) = { E(x) F (x) /(E(x) F (x)) if E(x) F (x), otherwise b E(x) F (x) dx = E = F E, F H 2π g(x) = e inx E n = F n H E = F 4.4 (). f f H 2π = L 2 (, 2π) (i) f f n f n = inf n f (x) = n Z inf n e inx. (ii) (iii) f n < +. n Z f(x) = f n e inx n Z x Proof. (i) g(x) = e inx 15

16 (ii) f f n f f n 2 < + n g(x) = e inx f n = inf n f n = 1 n f n n n n (iii) f(x) = lim 1 n 2 r 1 n Z 1/2 f n 2 n f n r n e inx 1/2 < + f(x) f n e inx n Z f(x) f n r n e inx n Z + f n (1 r n ) n Z 4.5. f(x) = x ( π x π). xe inx dx = i n xe inx + 1 n 2 e inx f = π/2 x = π 2 2 π n:odd 1 π x e inx dx = ( 1)n 1 2π π πn 2 (n ) 1 n 2 einx = π 2 4 (cos x + 13 π 2 cos(3x) + 15 ) 2 cos(5x) + x π x πz 27. * 2π f(x) f(x) = x ( π < x π) (ii) /π f f(x) = x sin(1/x) ( 2/π x 2/π) f 3. * f(x) = x α ( π x π) α > { 2α 2 f (x) 2 π 2α 1 dx = 2α 1 if 2α 1 >, + otherwise. 16

17 4.6. f(x + 2π) = f(x) {f n } f m f (m) n 2m+2 f n 2 < + n= k m f (k) (x) = (in) k f n e inx n= x Proof. (f (k) ) n = (in) k f n (k = 1, 2,, m) m = n 2 f n 2 < n Z f H 2π H 2π G(x) = inf n e inx n Z H 2π g(x) ing n = inf n g n = f n (n ) g = f 2π (f(x) g(x))e inx dx = ( n Z) f g h(x) = N k= N h ke ikx 2π f(x) g(x) 2 dx = f = g 2π 2π (f(x) g(x))(f(x) g(x) h(x)) dx f g h f(x) g(x) dx 4.7. (i) f m f (m) ( ) 1 f n = o n m (ii) f f n ( ) 1 f n = O n m+2 f m f (m)

18 fourier22 x = x = f ϵ >, +ϵ ϵ f (x) 2 < x = f(x) ( ϵ, ), (, + ϵ) x = f 4.8 (Dirichlet). f(x) x = lim n n k= n f k e ik = f( + ) + f( ) 2 f() Remrk. Gibbs 5 D R 2 D (Dirichlet problem) D f u(x, y) ((x, y) D) (x, y) D p D lim u(x, y) = f(p) (x,y) p u(x, y) ( ) 2 x y 2 u(x, y) = ((x, y) D), (men vlue property) (, b) D r > C B D u(, b) u u(, b) = 1 2π 2π u( + r cos θ, b + r sin θ) dθ. 18

19 r d dr 2π u( + r cos θ, b + r sin θ) dθ = = 2π 2π = 1 r = 1 r C B d u( + r cos θ, b + r sin θ) dθ dr (u x cos θ + u y sin θ) dθ (u x dy u y dx) (u xx + u yy ) dxdy =. D 32. D D = {(x, y) R 2 ; x 2 + y 2 < 1} f Poisson (x, y) z = x + iy ( ) 2 x y 2 = 4 z z = 4 z z f(e iθ ) θ f n z < 1 u(re iθ ) = f n r n e inθ = f n z n + f n z n n Z f(e iθ ) = n= lim r 1 u(reiθ ) θ ( 3.3) 33. * f n = 1 (n Z) u n=1 u t = D 2 u x 2. u(t, x) t D > 19

20 x q(t, x) = K u (t, x) x K > t x E(t, x) E(t + t, x) x E(t, x) x = q(t, x) t q(t, x + x) t, i.e., E = c u = c(u(t + t, x) u(t, x)) c > ( ) u(t + t, x) u(t, x)) u u c = K (t, x + x) (t, x) / x t x x t, x D = K/c R u u(t, x + 2πR) = u(t, x) x u(t, x) = u n (t)e inx/r n Z u n (t) = 1 2πR u(t, x)e inx/r dx u C 1 u t = u n(t)e inx/r n Z C 2 2 u x 2 = ( ) 2 in u n (t) e inx/r R n Z u n du n dt = D R n2 u n u n (t) = e tdn2 /R u n () f(x) u(t, x) = e tdn2 /R+inx/R f n, f n = 1 f(x)e inx/r dx 2πR n Z x = (f n = constnt, n Z ) (thet function) 2

21 34. * f t > x u(t, x) f(t) u(t, x) = u n (x)e int n Z 2π lim sup u(t, x) M. x u n (x) 1 2π 2π u(t, x) dt M u(t, ) = f(t) inu n (x) = Du n(x), u n () = f n. u n (x) = n e in/dx + b n e in/dx (n, n + b n = f n ), u (x) = f + cx. c = n=1 in/d = { (1 + i) n/2d if n 1, (1 i) n /2D if n 1 u(t, x) = f + f n e x n/2d e i(nt x n/2d) + f n e x n/2d e i(nt x n/2d). f(t) = + b sin t n=1 u(t, x) = + be x/ 2D sin(t x/ 2D). 21

22 x x/ 2D (phse *5 ) x = π 2D e π.4 3 C 1.2 C π 2D Wine cellr 6 ξ f(ξ) = + f(x)e ixξ dx *6 f (Fourier trnsform) 2L f(x) [ L, L] f (support) [ L, L] 2π F F (x) = f ( ) L π x, π x π F n = 1 π F (x)e inx dx = 1 f(x)e iπnx/l dx 2π π 2L L F n 2LF n n ξ = πn/l f F (y) = n F ne iny f f f(x) = n 1 2L f ( πn ) e ixπn/l L L f(x) = 1 e ixξ f(ξ) dξ 2π *7 Prsevl π π F (x) 2 dx = 2π n F n 2 *5 phse (topology) *6 1/2π *7 1/2π 22

23 L L L f(x) 2 dx = 1 π 2π L f(πn/l) 2 n f(x) 2 dx = 1 2π f(ξ) 2 dξ R L 1 (R) f L 1 (R) f f 1 = f(x) dx R (i) f 1 f 1 = f =, (ii) λf 1 = λ f 1, (iii) f + g 1 f 1 + g 1 (norm) (normed spce) L 1 (R) 6.1. f, g f g 1 f g L 1 (R) f g 1 [,b] [ ϵ, b + ϵ] C L f(x) lim f(x) x f L 1 (R) f(ξ) = R f(x)e ixξ dx ξ f(ξ) f 1 ξ n ξ lim dx = n R f(x)e ixξn R lim dx = n f(x)e ixξn R f(x)e ixξ dx R C b (R) C b (R) f = sup{ f(x) ; x R} {f n } C b (R) f C b (R) lim n f n f = f n (x) f(x) x R 23

24 6.2. f f L 1 (R) C b (R) f f f(x) = { e λx if x >, otherwise (λ > ) e λx ixξ dx = 1 λ + iξ. 1 + e ixξ 2π λ + iξ dξ f(ξ) 2 dξ = 6.4. f = 1 [,b] 1 λ 2 + ξ 2 dξ = π λ = 2π e 2λx dx = 2π f(x) 2 dx f(ξ) = b [ e e ixξ ixξ dx = iξ ] b = e iξ e ibξ iξ = O(1/ ξ ). f 36. f 6.5. e x2 ixξ dx = π 2 e ξ /4. (Gussin integrl) F (ξ) df dξ = ξ 2 F Cuchy ξ = f(x) = e x2 f(ξ) = π/e ξ2 /4 1 2π e ixξ f(ξ) dξ = e x 2 = f(x), 37. f(ξ) 2π 2 dξ = π = 2π f(x) 2 dx e x2 dx = π. 24

25 y = x + ξ 38. > f(x) = (x )e x2 f(x) dx < ( ). g(x) = f(x ) ĝ(ξ) = e ixξ f(x ) dx = e iξ f(ξ) h(x) = e ixη f(x) ĥ(ξ) = e ix(ξ η) f(x) dx = f(ξ η). e ixη F (x) (x R) (loclly integrble) [, b] F (t) dt < [,b] 39. f(x) F f(x) = F (t) dt + f(), [,x] x x [, b] C 1 g(t) b g (x)f(x) dx = b = [,b] = = ( ) g (x) F (y) dy + f() dx [,x] ( ) b F (y) g (x) dx dy + f()(g(b) g()) [,b] [,b] y F (y)g(y) dy + g(b) [,b] F (y)g(y) dy + f(b)g(b) f()g() F (y) + f()(g(b) g()) F f F = f f F 4. * F (x) = 1 x+i 25

26 6.7. F F L 1 (R) f lim f(x), x lim f(x) x Proof. f(x) = { [,x] F (t) dt if x, F (t) dt if x, [ x,] F g F b f(b) f() = f(x) g(x) [,b] F (t) dt = g(b) g() 41. f lim x ± f(x) f L 1 (R) 6.8. f, f L 1 (R) iξ f(ξ) = f (ξ), ξ R. Proof. f, f L 1 (R) f lim x ± f(x) f iξ b e ixξ f(x) dx =, b lim f(x) = x ± [,b] e iyξ f (y) dy + e iξ f() e ibξ f(b) iξ f(ξ) = iξ e ixξ f(x) dx = R e iyξ f (y) dy = f (ξ) f, f, f L 1 (R) f 42. * lim f(x) = x ± 6.1. [, ] f (, ), (, b), (, ) f f (ξ) = b e ixξ dx b e ixξ dx = 2b cos(ξ) 1 iξ 26

27 f(ξ) = f (ξ) iξ = 2b (1 cos(ξ) ξ f(ξ) ξ = 7 f 1 < *8 f 7.1. f(x) f(x) dx < f(ξ) f(ξ) dξ < f(x) = 1 e ixξ f(ξ) dξ 2π Proof. f (regulriztion) e ixξ f(ξ) dξ = lim e ixξ ξ2 f(ξ) dξ + e ξ2 +ixξ f(ξ) dξ = π = π = dy f(y) h(y) = f(x + y) 7.2. h(y) dξ e ξ2 +i(x y)ξ f(y)e (x y)2 /4 dy f(x + y)e y2 /4 dy. π lim h(y)e y2 /4 dy = 2πh(). + *8 C 2 f f, f, f ( 6.9) 27

28 Proof. h h h() h() = h(y) y = ϵ >, δ >, y δ = f(y) ϵ. h(y)e y2 /4 dy = y δ h(y)e y2 /4 dy + h(y)e y2 /4 dy y δ π δ h(y)e y2 /4 dy δ π h(y)e y2 /4 dy y δ δ/2 e x2 dx = π δ π ϵ e y2 /4 dy δ ϵ e y2 /4 dy = 2πϵ π h e y2 /4 dy = 4 π h e x2 dx y δ δ/2 e x2 dx * φ(ξ) R lim + δ/2 π 2 h(y)e y /4 = 2 φ(ξ) dξ = lim + R e ξ φ(ξ) dξ 7.3. < b λ > e λ(x b) if x b, f λ (x) = 1 if x b, e λ(x ) if x e x2 dx ( +) f λ (ξ) = λ λ ξ 2 iλξ e ibξ + ξ 2 + iλξ e iξ = O(1/ξ 2 ) 45. f ξ = L 2 (R) R R f(x) 2 dx < 28

29 L 2 (R) (f g) = f(x)g(x) dx R f 2 = (f f) f π(f f) = π f(x) 2 dx f x x L 2 (R) f /n h n h n /n λ n (x) λ n 1 = h n /n π(λ n λ n ) = π 2h2 n 3n lim λ n 1 = h n = o(n), n lim λ n 2 = h n = o(n 1/2 ). n Remrk. L 2 L 1 L 1 L R f(x) f(x) = O(1/x) f L 2 (R) f(x) = O(1/x 2 ) f L 1 (R) L 1 (R) L 2 (R) 47. * L 1 (R) \ L 2 (R), L 2 (R) \ L 1 (R) L 1 (R) \ L 2 (R) 7.5. (V, ) D V D V (dense) D V v V ϵ > v v ϵ v D 7.6. (i) 1 [,b] E E L 1 (R) L 2 (R) E L 1 (R) L 2 (R) (ii) C c (R) L 1 (R) L 2 (R) L 1 (R) L 2 (R) 29

30 8 Prsevl L 2 f f 1 f(x)f(x) dx = f(x) e ixξ 1 f(ξ) dξdx = f(x)e R 2π 2π ixξ dx f(ξ) dξ = 1 2π R f(ξ) 2 dξ f L 2 (R) 8.1. V W T : V W (unitry mp) V = W (unitry trnsformtion) (unitry opertor) 8.2 (Polriztion Identity). (v w) = 1 4 ( v + w 2 v w 2 i v + iw 2 + i v iw 2 ) f(x) L 1 (R) L 2 (R) f(x) dx < +, R f R f(x) 2 dx < + f(ξ) 2 dξ = 2π R f(x) 2 dx < + f f/ 2π L 2 (R) L 2 (R) f(x) = 1 f(ξ)e ixξ dξ 2π R 1 ( ) g(x)f(x) dx = f(ξ) 2π R g(x)eixξ dx dξ, g L 1 (R) L 2 (R) Proof. Gussin regulriztion lim e ixξ ξ2 f(ξ) dξ + 3

31 e ξ2 f(ξ) 2 dξ = dxdξ f(x)e ixξ ξ2 f(ξ) = dxdy f(x)f(y) dξ e ξ2 +i(x y)ξ π = dxdy f(x)f(y)e (x y)2 /4 π = dxdy f(x)f(x y)e y2 /4 π = 2 R h(y)e y /4 dy. h(y) = f(x)f(x y) dx R 7.2 f(ξ) π 2 dξ = lim h(y)e y2 /4 dy = 2πh() = 2π + R f(x) 2 dx. f L 2 (R) f(ξ)ĝ(ξ) dξ = 2π f(x)g(x) dx R 8.4. f, g L 2 (R) f g (f g)(y) = f(x)g(y x) dx R f g f 2 g 2. Proof. ( f(x y)g(y) dy ) 1/2 ( f(x y) 2 dy g(y) dy) 1/2 = f 2 g 2 f g f ϵ, g ϵ C c (R) f f ϵ 2 ϵ, g g ϵ 2 ϵ f g f ϵ g ϵ f f ϵ 2 g 2 + f ϵ 2 g g ϵ 2 g 2 ϵ + ( f 2 + ϵ)ϵ f ϵ g ϵ C c (R) f g *9 *9 lim x ± (f g)(x) = 31

32 48 (Riemnn-Lebesgue). f L 1 (R) lim f(ξ) = g, g L 1 (R) ξ ± lim x ± g(x) = f L f, g C c (R) f g C c (R) [f g] [f] + [g] 5. f, g L 1 (R) (x, y) f(x)g(y) R 2 f(x)g(y) dxdy = f(t)g(s t) dsdt. 2 2 R R s R f(t) g(s t) dt < R (f g)(s) = f(t)g(s t) dt R (f g)(s) ds = R R 51. f f(ξ) = f( ξ). f(x) dx g(y) dy R f f f(ξ) (ξ ) f(ξ) 2 ξ > power spectrum 8.5. V V V (Bnch spce) V v = (v v) 8.6. V D W Φ : D W Φ V Φ : V W Φ E = {Φ(v); v D} W Φ Proof. Φ v V v v n D Φ(v) Φ(v n ) = Φ(v v n ) = v v n (n ) Φ(v) = lim n Φ(v n) Φ Φ Φ : V W Φ(v) = lim n Φ(v n ) = lim n v n = v. 32

33 Φ : D E Ψ : E D E W Ψ : W V Ψ Φ(v) = v, Φ Ψ(w) = w, v D, w E V, W Φ L 2 (R) L 2 (R) 8.7 (Riesz-Fischer). (X, µ) L 2 (X, µ) lim n f n f 2 = (f n, f L 2 (X, µ)) n lim f n (x) = f(x) n for µ-.e. x X Proof. L 2 (X, µ) {f n } * 1 f n+1 f n 2 1/2 n ( n 2 f k+1 (x) f k (x) ) µ(dx) 1/2 X k=1 k=1 n f k+1 f k 2 1 n ( 2 f k+1 (x) f k (x) ) µ(dx) 1 X k=1 f k+1 (x) f k (x) < for µ-.e. x. k=1 f(x) = f 1 (x) + (f k+1 (x) f k (x)) k=1 x X (f k+1 (x) f k (x)) L 2 (X, µ) f L 2 (X, µ) f(x) f n (x) f k+1 (x) f k (x) k=1 k=n *1 {N k } k 1 m, n N k = f m f n 2 1/2 k 33

34 f f n 2 f k+1 f k 2 k=n k=n 1 s n. 2k 52. * L 1 (X, µ) L 1 (R) L 2 (R) E = { f; f L 1 (R) L 2 (R)} L 2 (R) Proof. 7.3 f λ f λ L 1 (R) L 2 (R) 7.1 2πf λ (x) = e ixξ fλ (ξ) f λ ( x) E lim λ f λ = 1 [,b] in L 2 (R) 1 [ b, ] E E E E L 2 (R) L 1 (R) L 2 (R) f f L 2 (R) L 2 (R) L 2 (R) L 2 f f L 2 (R) L 2 f L 2 (R) R ĝ(ξ) f(ξ) dξ = 2π g(x)f(x) dx, R g L1 (R) L 2 (R) h L 2 (R) ĝ(ξ)h(ξ) dξ = 2π R g(x)f(x) dx, R g L1 (R) L 2 (R) h = f 53. f(x) = { e λx if x >, otherwise (λ > ) f(ξ) = 1 + 2π e λx ixξ dx = 1 λ + iξ e ixξ λ + iξ dξ = { e λx if x >, if x < g L 1 (R) L 2 (R) 1 ( e ixξ ) g(x) dx dξ = e λx g(x) dx. 2π R λ + iξ (, ) 34

35 L 2 g h L 2 (R) h(x)g(x) dx R h(x) h(x) h h g probe > [, ) g r lim dξ r r x * 11 r r x π dθ r r iλe iθ e rx sin θ 2 e ixξ r g(x) dx = lim dx g(x) λ + iξ r x r e ixξ λ + iξ dξ = 2πe λx + π/2 dθ π r r λ e rx2θ/π = e ixξ λ + iξ dξ r cos θ xr sin θ dθ eixr r iλe iθ π (r λ)x (1 e rx ) r e ixξ lim dx g(x) dξ = 2π e λx g(x) dx = 2π r x r λ + iξ x R e λx g(x) dx e λx (x > ) 54. * x < π (r λ)x π (r λ) { x λ e x if x >, f(x) = otherwise. > λ x λ = e λ log x, x >. f(ξ) = + x z = ( + iξ)x f(ξ) = ( + iξ) λ 1 x λ e x ixξ dx L z λ e z dz *11 (Fubini ) 35

36 L + iξ e z + f(ξ) = ( + iξ) λ 1 x λ e x dx = ( + iξ) λ 1 Γ(λ + 1) ( + iξ) λ 1 e ixξ dξ = 55. { 2πx λ e x /Γ(λ + 1) if x >, otherwise Prsevl Dirichlet fourier (Dirichlet). f L 1 (R) L 2 (R) x = f( + ) + f( ) 2 N = lim N N f(ξ)e iξ dξ 9 g(x) = 1 2π x ξ g(x) = dξ g(y)δ(x y) dy = dy g(y)e i(x y)ξ δ(x) = 1 e ixξ dξ 2π f(y) = g( y) f(y)δ(y) dy = f() g(x y)δ(y) dy, δ(y) f(y) y = +ϵ ϵ f(y)δ(y) dy = f() = 36

37 δ() = ( ) f(y) = 1 ( y ϵ) δ(y) dy = f(y)δ(y) dy = f() = 1 δ(y) 1 y = δ(x) 1 2π e ξ2 +ixξ dξ = 1 4π e x2 /4 + 1 dξe ixξ = sin(x) 2π πx + wild δ(x) φ (x) δ(x) = lim φ (x) {φ } (i) φ (x) (ii) f(x) lim f(x)φ (x) dx = f() φ φ(x) f φ f φ(x)f(x)dx φ(x)f(y)dx (liner functionl) δ f f() 37

38 φ δ = lim φ + lim φ (x)f(x) dx = f() + φ g (gφ)(f) = g(x)φ(x)f(x) dx = φ(gf) φ(x) ψ(x) = φ(x ) ψ(f) = φ(x )f(x) dx = φ(x)f(x + ) dx = φ(f ), f (x) = f(x + ) ψ = φ φ φ ( ) ( ) φ φ (f) φ(f) f f f f (f) = lim = lim φ = φ lim = φ(f ) φ f f D = {f C (R); f } S = {f C (R); m, n, x m f (n) (x) } D φ (distribution) D n < b, C >, n N, f D, [f] [, b] = φ(f) C f (k) S (tempered distribution) S * 12 C >, m, n N, f S, φ(f) C mx{ x k f (l) (x) ; x R}. k m,l n *12 prime str k= 38

39 D S S D D Remrk. D S D S D S [Reed-Simon] φ T φ (f) = φ(x)f(x) dx, R f D T φ : D C 9.1. φ, ψ T φ = T ψ φ(x) = ψ(x) for.e. x R Proof. φ T φ ψ = > b > h = { φ(x) if x nd φ(x) b, otherwise [f] [, ] f b f D T φ (f) =, b [,] [ φ b] φ T φ φ(x) 2 dx = 9.2. Heviside function h { 1 if x, h(x) = otherwise f D h(x)f (x) dx = f (x) dx = f() h = δ * C h(x) h(x) = (x ) h(x) = 1 (x 1) (i) C c (R) = Ch + (C c (R)) (ii) φ = φ : C c (R) C φ(h ) φ(f) = φ(h ) f(x) dx, f Cc (R). 39

40 9.3. f (tempered) f(x) R (1 + x 2 ) n dx < n f fg (g S) S g f(x)g(x) dx R T f Remrk. f T f D S f : R R e if(x) T f T e if = it f e if T f e if S f f e if Remrk. h D h 1, h(x) = 1 for x 1 h n D (n = 1, 2,... ) { 1 if x n, h n (x) = h( x n + 1 ) otherwise S h n g g (g S) f T f D g S, lim n R f(x)g(x)h n(x) dx T : S C T : S C ( T )(f) = T ( f), f S 58. S 9.6. (i) f L 1 (R) T f T f (ii) f L 2 (R) f L 2 f T f = T f πδ(x) 9.8. T : S C T = 2π T. T T (g) = T (x)g( x) dx R 4

41 f (f, f L 1 (R)) f(x) = 1 f(ξ)e ixξ dξ 2π R 2πg = f Cb (R) for.e. x R f(x) = g( x) for.e. x R 59. f, f L 1 (R) f L 2 (R) 2πT g = T f = Tf = 2π T f = 2πT f. 1 (wve eqution) u(, x) = f(x), 2 u t 2 2 u x 2 = u (, x) = g(x) t u(t, x) = 1 v(t, ξ)e ixξ dξ 2π v(, ξ) = v(, ξ) = v(t, ξ) = iξ f(ξ) + ĝ(ξ) 2iξ u(t, x) = g h u(t, x) = f(x)e ixξ dx, g(x)e ixξ dx, 2 v t 2 (t, ξ) = ξ2 v(t, ξ) f(x + t) + f(x t) 2 f(x + t) + h(x + t) 2 e itξ + iξ f(ξ) ĝ(ξ) 2iξ d Alembert F (x + t) + G(x t) + x+t x t e itξ g(y) dy. f(x t) h(x t) 2 41

42 6. y = x + t, z = x t d Alembert (het eqution) u t = D 2 u x 2 (D > ) lim u(t, x) = f(x) t + u(t, x) = 1 e ixξ G(t, ξ)dξ 2π G(, ξ) = u(t, x) f(y)e iyξ dy. G t (t, ξ) = Dξ2 G(t, ξ) G(t, ξ) = e Dξ2t G(, ξ) = e Dξ2 t f(y)e iyξ dy. u(t, x) = 1 dy f(y) dξe Dtξ2 +i(x y)ξ dξ 2π 1 = e (x y)2 /(4Dt) f(y) dy 4πDt f(x) = δ(x) x = u(t, x) = 1 4πDt e x2 /(4Dt) * σ > f(x) = e (x )2 /2σ 2 u(t, x) Dirichlet y > (Lplce eqution) 2 u x u y 2 = 42

43 lim u(x, y) = f(x), lim y + u(x, y) = y + u(x, y) = 1 F (ξ, y)e ixξ dξ 2π (iξ) 2 F + 2 F y 2 = F (ξ, y) = A(ξ)e ξy + B(ξ)e ξy F (ξ, y) = f(ξ)e y ξ u(x, y) = 1 2π = y π f(ξ)e ξ y e ixξ dξ = 1 2π f(t) (x t) 2 + y 2 dt dtdξ f(t)e y ξ e i(x t)ξ 63. f(x) u(x, y) 11 C V V φ : V C φ(v + w) = φ(v) + φ(w), φ(λv) = λφ(v), v, w V, λ C * 13 (liner functionl) (i) V = C n φ : V C v 1 v =. ( v 1 ) φ 1... φ. n n = φ j v j φ 1,..., φ n C. (ii) V v V v v n v n j=1 v (v ) = (v v ), v V *13 (liner form) 43

44 V φ : V C (continuous) lim v n = v = lim φ(v n) = φ(v) n n V V V (φ + ψ)(v) = φ(v) + ψ(v), (λφ)(v) = λφ(v) V (dul spce) 64. R C c (R) L 2 (R) C c (R) f f() C V φ (bounded) * 14 { φ(v) ; v V, v 1} M > v 1 = φ(v) M V φ (i) φ (ii) φ (iii) φ 1 () = {v V ; φ(v) = } Proof. (ii) = (i) = (iii) (iii) = (ii): p(v) = inf{ v + x ; x φ 1 ()} p(λv) = λ p(v), p(v) v, p(v + x) = p(v), λ C, v V, x φ 1 () φ 1 () p(v) = v φ 1 () w V φ(w) = 1 V = φ 1 () + Cw v = x + λw p(w) > φ(v) = λ = λ p(w) p(w) = p(λw) p(w) = p(v) p(w) 1 p(w) v. φ φ = sup{ φ(v) ; v V, v 1} φ V *14 44

45 11.4. φ = inf{m > ; φ(v) M v for ny v V } V φ Proof. φ lim φ m φ n = m,n v V {φ n (v)} n 1 ( φ(v) = lim n φ n (v) φ n φ φ ϵ >, m, n N φ m (v) φ n (v) φ m φ n v ϵ v v m φ(v) φ n (v) ϵ v φ φ n n N, φ φ n ϵ φ = (φ φ n ) + φ n 66. * lim φ φ n = n V w (w V ) w = w. (w v) w v = w if v 1, w w w = w v = w/ w w (v) = (w v) = w w w 67. * h C b (R) L 1 (R) φ φ(f) = h(x)f(x) dx R f L1 (R) φ 45

46 Riesz * (F. Riesz). H φ : H C w H φ(v) = (w v), v H φ = w φ w φ = w Proof. H {e i } i I φ j = φ(e j ) F I {z j } j F φ j z j = φ( z j e j ) φ z j e j j F j F = φ z j 2 j F z j = φ j j F φ j 2 φ 2, φ j 2 φ 2 j I w H j F w = j I φ j e j φ(e j ) = φ j = (w e j ) = w (e j ), j I. φ, w j I Ce j φ = w w H w (w w v) = φ(v) φ(v) = v = w w w w = V {e j } j I V = l 2 (I) H H H v v H (i) (ii) (v w ) = (w v), v, w H H H v v v v (w ) = (v w ) = (w v) = w (v) *15 Riesz Riesz lemm 46

47 v v v v H = H V V V H = H 68. V V f L 2 (R), g L 1 (R) L 2 (R) f g L 2 (R) f g 2 f 2 g 1 h L 2 (R) f g(x)h(x) dx f(x y)g(y)h(x) dxdy ( dy g(y) ) 1/2 ( f(x y) 2 dx ) 1/2 h(x) 2 dx = f 2 h 2 g 1 f g L 2 (R) f g 2 f 2 g 1 von Neumnn Rdon-Nikodym elegnt hilbert T : V W (i) T (ii) T {v V ; v = 1} (iii) T {v V ; v v < r} T = sup{ T (v) ; v V, v = 1} T : V W S : W V (w T v) = (Sw v) ( v V, w W ) T (hermitin conjugte) S = T T : V W (i) T (ii) T 47

48 Proof. (ii) = (i) Riesz lemm (w T v) w T v w T v w W v (w T v) Riesz lemm v V (w T v) = (v v) (v V ) v w v = S(w) (w T v) w S(w) (i) = (ii) (i) T v = sup{ (w T v) ; w W, w = 1} = sup{ (Sw v) ; w W, w = 1} v (Sw v) v v V, ϵ >, δ >, v v δ = T v T v ϵ. T T ( ) Bolzno B r (v) = {v V ; v v < r}, B r (v) = {v V ; v v r} v 1 B 1 () T v 1 > 1 T ( ) < r 1 1/2 B r1 (v 1 ) B 1 () v v 1 r 1 = T v 1 v 2 B r1 (v 1 ) T v 2 > 2 < r 2 1/2 2 B r2 (v 2 ) B r1 (v 1 ) v v 2 r 2 = T v 2 B rn (v n ) B rn 1 (v n 1 ) r n 1/2 n v v n r n = T v n {v n } n 1 V v = lim n v n {v n } n m B rm (v m ) v B rm (v m ) T v m m 1 T v < 71. * (). H E v H v = w + w, w E, w E Proof. E E = {} E {e j } v H w = j v)e j j I(e E (v w e k ) = (v e k ) j (v e j )(e j e k ) =, k I v w E 48

49 E (E ) = E. Proof. E = (E) H = E + E Dirichlet L 2 R d Ω Ω C Cc (Ω) Cc (Ω) L 2 (Ω) K u = L 2 K K {} Ω f(x) L 2 f Ω F F K + K K u u L 2 K Ω u A L 1 L 2 (i) E E L 1 (R) L 2 (R) (hilbert212 Appendix B) X A (indictor function) 1 A 1 A (x) = 1 (x A), 1 A (x) = (x A) (ii) C n C n c (R) L 1 (R) = C n c (R) L1, L 2 (R) = Cc n (R) L2 Cc n (R) (iii) (iv) (X, µ) f n (x) g(x) f n (x) g(x) (x X, n 1) lim f n(x) = f(x), µ-.e.x X n lim f n (x) µ(dx) = f(x) µ(dx). n X X Fubini f(x, y) dxdy < 2 R f(x, y) dxdy = 2 R R ( ) f(x, y) dx R 49 ( dy = R f(x, y) dy R ) dx

50 f(x, y) dxdy = 2 R ( ) ( f(x, y) dx dy = R R R f(x, y) dy R ) dx B vs. f(x) f(x ) T (trnsltion) f(x), g(x) (convolution) lim T yf T f =. y (f g)(x) = f(x y)g(y)dy f g = g f, (f g) h = f (g h). f(x) f(rx) ( r R) f(x) rf(rx) f(x + ) f (n) (x) f(ξ) f(ξ/r) iξ e f(ξ) (iξ) n f(ξ) e iαx f(x) f(ξ + α) 2πf(x)g(x) f g f ĝ f(ξ)ĝ(ξ) prmeter Tylor 72. B.1 (Riemnn-Lebesgue). f(x) f(x) dx < + f(ξ) ξ lim f(ξ) =. ξ ± 5

51 Proof. f g f(x) g(x) dx f(ξ) f(ξ) ĝ(ξ) + ĝ(ξ) ĝ(ξ) (ξ ) f g ξ B.2. (i) f(x) m lim f (l) (x) = ( l m 1), x ± f (m) (x) dx < + f(ξ) ( ) 1 f(ξ) = o ξ m lim ξ ± ξm f(ξ) = (ii) f(ξ) f(ξ) = O(1/ ξ m+2 ) f m f (m) lim f (l) (x) = ( l m), x ± Proof. (i) f (l) f (m) (x)e ixξ dx = (iξ) m f(x)e ixξ dx (iξ) m f(ξ) (ii) f f(ξ) = O(1/ ξ 2 ) Riemnn-Lebesgue f(x) = 1 f(ξ)dξ 2π f(ξ) = O(1/ ξ 3 ) f f (x) = 1 2π (iξ) f(ξ)e ixξ dξ iξ f(ξ) Riemnn-Lebesgue f (x) 51

52 C Plncherel formul L 1 (R) L 2 (R) f f L 2 (R) L 2 (R) L 2 (R) R ĝ(ξ) f(ξ) dξ = 2π g(x)f(x) dx, R f, g L2 (R) Plncherel C.1. f L 2 (R) f L 1 (R) x R f(x) = 1 2π R eixξ f(ξ) dξ Proof. Plncherel formul g L 1 L 2 g, f l.h.s. = dξ f(ξ) R R dx g(x)e ixξ = dx g(x) R R dξ eixξ f(ξ) g L 1 L 2 R ( ) dx g(x) 2πf(x) R dξ eixξ f(ξ) = f L 2 (R) f L 2 (R) (t R) g t (x) = 1 2π t ĝ t = { 1 [,t] if t, 1 [t,] if t < e ixξ dξ = eitx 1 2πix Plncherel formul t f(ξ) dξ = R f t R L2 (R) \ L 1 (R). e itx 1 f(x) dx ix e itx 1 f(x) dx ix f(ξ) = d e ixξ 1 f(x) dx dξ R ix 73. g t g s 2 2 = 2π t s 52

53 D z = re iθ Re ( ) 1 + z = 1 r2 1 z 1 z 2 f(e it ) D = {z C; z < 1} φ(z) = 1 2π e it + z 2π e it z f(eit ) dt iα (α R) φ() R φ(z) = iim(φ()) + 1 2π e it + z 2π e it z f(eit ) dt, f(e iθ ) = lim r 1 Re(φ(reiθ )) φ(z) D D.1. D φ(z) Re(φ(z)) (z D) D = {z C; z = 1} µ φ(z) = iim(φ()) + 1 2π e it + z 2π e it z µ(dt) Proof. Re ( e it ) + z e it = 1 r2 z 1 e it z 2 = P r(θ t) z < ρ < 1 φ(z) = iim(φ()) + 1 2π ρe it + z 2π ρe it z Re(φ(ρeit )) dt µ ρ (dt) 1 2π µ ρ (dt) = Re(φ(ρe it )) dt 2π µ ρ (dt) = Re(φ()) ρ 1 1 {ρ n } µ = lim n µ ρ n 53

54 * 16 t ρ n e it + z lim n ρ n e it z = eit + z e it z Remrk. Re(φ(z)) = z D φ Re(φ) µ D z w = i 1 + z 1 z, z = w i w + i D {w C; Im(w) > } ψ(w) = iφ(z) ψ e it + z e it z = 1 w cot(t/2) 1 i w + cot(t/2) s = cot t 2, < t < 2π R λ µ(dt)/2π = λ(ds) ψ(w) = Im(φ()) + i 2π e it + z (,2π) e it z µ(dt) + i 2π = Re(ψ(i)) + µ({}) 2π w sw R s w λ(ds) 1 + sw = α + βw + R s w λ(ds) α R, β R ω 1 Ω(z) = R t z ω(dt) 1 + z 1 z µ({}) C \ R Ω ω Cuchy-Stieltjes trnsform D.2 (Stieltjes inversion formul). 2πiω(dt) = lim (Ω(t + iϵ) Ω(t iϵ)) dt. ϵ + D.3. Ω ω Cuchy-Stieltjes (i) Ω(z) = Ω(z) (z C \ R). (ii) Ω (iii) {y Ω(iy) ; y 1} *16 hilbert

55 Proof. (ii) 1 + tz Ω(z) = α + βw + λ(dt), Im(z) > R t z (i) z C \ R yω(iy) αy + y(1 y 2 ) (iii) t t t 2 + y 2 λ(dt), βy2 + y t 2 + y 2 λ(dt) (y 2 1)t β =, α = lim y + t 2 + y 2 λ(dt). T > y + y 2 t t 2 + y 2 λ(dt) M [ T,T ] [ T,T ] t t 2 + y 2 λ(dt) M (t 2 + 1) λ(dt) M T + (t 2 + 1) λ(dt) M λ t λ(dt) < α α = (y 2 1)t lim y + t 2 + y 2 λ(dt) = tλ(dt). Ω Ω(z) = R t λ(dt) + R ω(dt) = (t 2 + 1)λ(dt) 1 + tz t z λ(dt) = t R t z λ(dt) Remrk. ω Cuchy-Stieltjes Ω(z) lim yω(iy) =, lim y + yω(iy) = i y + R ω(dt). 55

56 D.4. t 1 < t 2 < < t 2n+1 G(z) = (z t 2)(z t 4 ) (z t 2n ) (z t 1 )(z t 3 ) (z t 2n+1 ) G(z) ω k = G(z) = n k= ω k z t 2k+1 lim (z t 2k+1 )G(z) = (t 2k+1 t 2 )... (t 2k+1 t 2k )(t 2k+1 t 2k+2 )... (t 2k+1 t 2n ) z t 2k+1 (t 2k+1 t 1 )... (t 2k+1 t 2k 1 )(t 2k+1 t 2k+3 )... (t 2k+1 t 2n+1 ) > G k ω kδ(t t 2k+1 ) Cuchy-Stieltjes 56