A Bowl of Kernels. By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty. December 03, 2013

Transcription

1 December 03, 203

2 Introduction When we studied Fourier series we improved convergence by convolving with the Fejer and Poisson kernels on T. The analogous Fejer and Poisson kernels on the real line help us improve convergence of the Fourier integral. In this project we define the Fejer kernel,the Poisson kernel, the heat kernel, the Dirichlet kernel, and the conjugate Poisson kernel. We study some of their properties, applications, and connections with complex variables.

3 Kernels Fejer Kernel on F pxq sin πx πx 2 0 () The Poisson kernel on P y pxq The Heat Kernel on y πpx 2 y 2 q y 0 (2) H t pxq e x 2 4t? t 0 (3) 4πt

4 Kernels The Dirichlet kernel on D pxq e 2πixξ dξ sinp2πxq πx The Conjugate Poisson kernel on 0 (4) Q y pxq x πpx 2 y 2 q y 0 (5)

5 Kernels Poisson Kernel and Conjugate Poisson Kernel solve the Laplace equation. P p B2 qp p B2 qp 0, Bx 2 By 2 Q p B2 qq p B2 qq 0 for px, yq in the upper half-plane Bx 2 By 2 px P, y 0q. Proof(Poisson Kernel): BP Bx y π p qpx 2 y 2 q 2 2x BP Bx 2 8x 2 y πpx 2 y 2 q 3 2y πpx 2 y 2 q 2 BP By x 2 y 2 πpx 2 y 2 q 2 BP By 2 6yx 2 2y 3 πpx 2 y 2 q 3

6 Kernels P 8x 2 y πpx 2 y 2 q 3 2y 6yx 2 2y 3 πpx 2 y 2 q 2 πpx 2 y 2 q 3 2x 2 y 2y 3 2ypx 2 y 2 q πpx 2 y 2 q 3 0 Theorem Heat Kernel is a solution of the heat equation on the line. p B B2 Bt qh p qh Bx 2

7 Approximation of The Identity In class we learned that an Approximation of the Identity on is a family tk t u tpλ where the following holds: The functions tk t u tpλ have mean value : ³ 8 8 K tpxqdx 2 The functions are uniformly integrable in t: There is a constant C 0 such that ³ 8 8 K tpxq dx C for all t P Λ 3 Concentration of mass at the origin: For each δ 0 ³ x δ K tpxq dx 0 lim tñt0

8 Fejer Kernel is an Approximation to the Identity ecall the Fejer Kernel is defined as, F pxq sin πx 2 0 (6) πx Mean value : sin πx 2 dx (7) πx Let u πx, π dx du π sin u π u 2 du π Then we have, cos 2 u u 2 du (8) p cos uqp cos uq u 2 du (9)

9 Fejer Kernel Now we can use integration by parts with u cos u and dv cos u du ³ πu 2 cos x Then we can use the well known identity, dx π and we x 2 have, p cos uq sin u du p cos uq cos u (0) Uniformly Integrable in t: ³ 8 There is a constant C 0 such that 8 sin πx 2 dx πx C for all 0 ³ But by, we know sin πx 2dx πx and also sin πx 2 πx 0 So just let C=.

10 Fejer Kernel Concentration of mass at the origin: For each δ 0 lim Ñ8 ³ x δ sin πx 2 dx πx 0. Let δ 0, We know 2 sin πx πx So we have, pπxq 2 pπxq 2 x δ sin πx πx Now we take the limit in, lim LÑ8 pπq 2 L pπq 2 δ lim Ñ8 2 pπq 2 δ δ pπxq 2 () (2) (3) 2 pπq 2 δ 0 (4)

11 Continuous Functions of Moderate Decrease Definition A function f is said to be a continuous function of moderate decrease if f is continuous and if there are constants A and ɛ 0 such that A f pxq p x ɛ q for all x P

12 Theorem Neither the Poisson kernel nor the Fejer kernel belong to S(). For each 0 and each y 0 the Fejer kernel F and the Poisson kernel P y are continuous functions of moderate decrease with ɛ. The conjugate Poisson kernel Q y is not a continuous function of moderate decrease. Proof(For Poisson Kernel) P y pxq If 0 y, y πpx 2 y 2 q for y 0 y P y pxq πy 2 pp x y q2 q pπyqp x 2 y 2 q

13 If 0 y then y 2 y 2 x 2 y 2 x 2 x 2 y 2 x 2 Then P y pxq x pπyqp 2 q y 2 πypx 2 q

14 If y y y 2 x 2 y 2 gleqx 2 Then Hence, y P y pxq πy 2 pp x y q2 q A y y πpx 2 q # πy if 0 y, y π if y

15 Kernels Theorem Let S f denote the partial Fourier integral of f, defined by S f pxq ˆf pξqe 2πixξ dξ (5) ³ 0 D tpxqdt. Then S f pxq D f and F pxq Proof: D f D px yqf pyqdy (6) e 2πpx yqξ dξf pyqdy (7) e 2πiyξ f pyqdye 2πixξ dξ ˆ f pξqe 2πixξ dξ (8) (9)

16 Kernels Also, F ³ pxq 0 D tpxqdt because, 0 D t pxqdt 0 πx sinp2πtxq dt (20) πx cosp2πtxq 2πx 0 (2) 2π 2 pcosp0q cosp2πxqq (22) x 2 psinpπxq2 π 2 x 2 F pxq (23)

17 Integral Cesaro Means We have the Fejer Kernel can be written as the integral mean of the Dirichlet kernel, and the Fejer kernel define an approximation to the identity on as Ñ 8. Therefore, the integral Cesaro means of a function of moderate decrease converge to f as Ñ 8 Theorem σ f pxq The convergence is both uniform and in L p 0 S t f pxqdt F f pxq Ñ f pxq (24)

18 Integral Cesaro Means We will prove for p=.we will show, lim F f f 0 (25) Ñ8 F pxq f pxq f pxq F f f f px yqf pyqdy f pxq (26) px yq f pxqqf pyqdy pf (27) pf px yq f pxqq F pyq dy (28) pf px yq f pxqq F pyq dydx (29) (30)

19 Integral Cesaro Means By Fubini, F f f y δ y δ pf px yq f pxqq F pyq dy (3) pf px yq f pxqq F pyq dy (32) pf px yq f pxqq F pyq dy (33) I I 2 (34) Notice, I 2 ³ y δ pf px yq f pxqq F pyq dy can be made arbitrarily small by continuity of f.

20 Integral Cesaro Means Also, I ³ y δ pf px yq f pxqq F pyq dy can be made arbitrarily small since F is an approximation to the identity. This concludes the proof, and we have σ f pxq 0 S t f pxqdt F f pxq Ñ f pxq (35)

21 Fourier Transform of the Fejer Kernel Since the Fejer Kernel is a continuous function of moderate decrease, we can actualy calculate its Fourier Transform. Let f be any function with a Fourier Transform supported on and ˆf pξq 0 F f 0 0 t 0 D t pxqdt f (36) pd t pxq f qdt (37) t ˆf pξqe 2πixξ dξ ˆf pξqe 2πixξ ξ dt (38) dt dξ (39)

22 Fourier Transform of the Fejer Kernel F f ˆf pξqe 2πixξ ξ dξ (40) 8 ˆf pξq ξ χ 8 r,s pξqe 2πixξ dξ (4) ˆf pξq ξ χ r,s pξq q (42) Now we can take the transform, {F f ˆf pξq ξ {F pξqf pξq ˆ ˆf pξq ξ {F pξq ξ χ r,s pξq (43) χ r,s pξq (44) χ r,s pξq (45)

23 Connection with Complex Variables Consider a real valued function f P L 2 pq. Let F pz) be twice the analytic extension of f to the upper half plane 2 tz x iy : y 0u Then F pzq is given explicitly by the well known Cauchy Integral Formula: F pzq πi f ptq dt (46) t z If we separate F pzq into its real and imaginary parts, we get F pzq πi f ptq dt (47) pt xq iy

24 Connection with Complex Variables F pzq f ptqppt xq iyq πi pt xq 2 y 2 dt (48) i f ptqpx tq π pt xq 2 y 2 dt f ptqy π pt xq 2 dt (49) y 2 f P y pxq ipf Q y pxqq (50) u iv The function u is called the harmonic extension of f to the upper half plane, while v is called the harmonic conjugate of u. (5)

25 Theorem Q y L pq, but Q y P L 2 pq Proof x 0 πpx 2 y 2 q dx x 8 8 πpx 2 y 2 q dx 0 8 u du y 2 2π u du 8 2π u du 2π u du π y 2 8 2π 8 y 2 8 y 2 u du y 2 x πpx 2 y 2 q π ln u 8 y 2 8 So Q y pxq L pq

26 Connection with Complex Variables The conjugate Poisson Kernel is not in L pq and is not a function of moderate decrease, so we define its Fourier transform as a principal value: ˆ Q y pξq lim Ñ8 lim Ñ8 e 2πiξx e 2πiξx Q y pxqdx (52) x πpx 2 y 2 dx (53) q Now, consider the function f pzq e 2πiξz πpz 2 y 2 q and apply residue theorem for a well chosen contour in the complex plane. z (54)

27 connection to complex variables For ξ 0 we choose the semi circle in the upper half plane and for ξ 0 we choose the semi circle in the lower half plane. Now, lets find the residues of f at iy and iy. espf pzq, iyq e 2πiξpiyq 2π e2πiξy 2π (55) for ξ o (56) espf pzq, iyq e 2πiξp iyq 2π e 2πiξy 2π (57) for ξ 0 (58)

28 Connection with complex variables Therefore ˆ Q y pξq isgnpξqe 2πy ξ (59) Definition The Hilbert Transform H is defined by : Hf pxq π lim ɛñ0 ³ x y ɛ Definition ˆ Hf pξq isgnpξqˆf pξq f pyq x y dy for f P L2 pq We can see that the Hilbert transform can be written as the principal value of the convolution of f and πx. It is the response to f of a linear time invariant filter(hilbert Transformer) having impulse response πx

29 Connection with Complex Variables The Hilbert transform find a harmonic conjugate yptq for a real function xptq so that zptq xptq iyptq can be analytically extended from to the upper half plane.in signal processing, the HT can be interpreted as a way to represent a narrow band signal in terms of amplitude and frequency modulation.in the Fourier side, we can think of the HT as a phase shifter by 90 degrees. If we take the limit of Q y ˆpξq as y goes to zero, we get ˆ Q y pξq Ñ isgnpξq (60) Then as y Ñ 0, Q y pxq f approaches the Hilbert transform Hf in L 2 pq

30 Applications Consider the initial value problem for the heat equation: u t u xx 0 in xp0, 8q with u g on xpt 0q If we take the Fourier transform of u in x we get: û t y 2 û 0 for t 0 and û ĝ for t 0 Consequently, û e ty 2 ĝ and u e ty 2 ĝ therefore u g F where ˆF e ty 2. But then p2πq 2 F pe ty 2 q (6) e ixy ty 2 dy (62) p2πq 2 Therefore, upx, tq p4πtq 2 2t e x2 4t (63) e x y 2 gpyqdy x P, t 0 (64)

31 eferences Harmonic Analysis:from Fourier to Wavelets. Cristina Pereyra and Lesly Ward. AMS, 200. Partial Differential Equations. Lawrance Evans.AMS, 998. Fourier Analysis. Stein and Shakarchi.Princetton University Press, 2003.