A Bowl of Kernels

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty

December 03, 2013

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

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Kernels

Fejer Kernel on R ¢ sin πRx 2 F pxq R R ¡ 0 (1) R πRx

The Poisson kernel on R y P pxq y ¡ 0 (2) y πpx2 y 2q The Heat Kernel on R

¡|x|2 e 4t Ht pxq ? t ¡ 0 (3) 4πt

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Kernels

The Dirichlet kernel on R » R p q 2πixξ sin 2πRx DR pxq e dξ R ¡ 0 (4) ¡R πx The Conjugate Poisson kernel on R x Q pxq y ¡ 0 (5) y πpx2 y 2q

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Kernels

Poisson Kernel and Conjugate Poisson Kernel solve the Laplace p B2 q p B2 q equation. ∆P Bx2 P By 2 P 0, p B2 q p B2 q p q ∆Q Bx2 Q By 2 Q 0 for x, y in the upper half-plane px P R, y ¡ 0q. Proof(Poisson Kernel):

BP y p¡1qpx2 y 2q¡22x Bx π B 2 P 8x y ¡ 2y Bx2 πpx2 y 2q3 πpx2 y 2q2 B 2 ¡ 2 P x y By πpx2 y 2q2 B ¡ 2 3 P 6yx 2y By 2 πpx2 y 2q3

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Kernels

8x2y 2y ¡6yx2 2y 3 ∆P ¡ πpx2 y 2q3 πpx2 y 2q2 πpx2 y 2q3 2 3 ¡ p 2 2q 2x y 2y 2y x y πpx2 y 2q3 0

Theorem Heat Kernel is a solution of the heat equation on the line. p B q p B2 q Bt H Bx2 H

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Approximation of The Identity

In class we learned that an Approximation of the Identity on R is a t u family Kt tPΛ where the following holds: ³8 1 t u p q The functions Kt tPΛ have mean value 1: ¡8 Kt x dx 1 2 The functions are uniformly integrable in t: There is a ³8 constant C ¥ 0 such that ¡8 |Kt pxq|dx ¤ C for all t P Λ 3 Concentration of³ mass at the origin: For each ¡ Ñ | p q| δ 0 limt t0 |x|¥δ Kt x dx 0

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Fejer Kernel is an Approximation to the Identity

Recall the Fejer Kernel is deﬁned as, ¢ sin πRx 2 F pxq R R ¡ 0 (6) R πRx Mean value 1 : » ¢ sin πRx 2 R dx 1 (7) R πRx du Let u πRx, dx πR Then we have, » ¢ » 2 ¡ 2 1 sin u 1 1 cos u du 2 du (8) π R u π R u » p ¡ qp q 1 1 cos u 1 cos u 2 du (9) π R u

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Fejer Kernel

Now we can use integration by parts with u1 1 cos u and 1¡cos u 1 dv πu2 du ³ 1¡cos x Then we can use the well known identity, R x2 dx π and we have,

» p1 cos uq sin u1du1 p1 cos uq ¡ cos u 1 (10) R

Uniformly Integrable in t: ³ ¨ ¥ 8 | sin πRx 2| ¤ There is a constant C 0 such that ¡8 R πRx dx C for all R ¡ 0 ³ ¨ ¨ sin πRx 2 sin πRx 2¥ But by 1, we know R R πRx dx 1 and also R πRx 0 So just let C=1.

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Fejer Kernel

Concentration of³ mass at the origin:¨ For each ¡ | sin πRx 2| δ 0 limRÑ8 |x|¥δ R πRx dx 0. Let δ ¡ 0, We know¨ sin πRx 2 ¤ 1 1 R πRx R pπRxq2 Rpπxq2 So we have, ¢ » 2 » 8 sin πRx ¤ 1 R 2 2 (11) |x|¥δ πRx δ Rpπxq 1 1 2 lim ¡ (12) LÑ8 Rpπq2L Rpπq2δ 2 (13) Rpπq2δ Now we take the limit in R, 2 lim 0 (14) RÑ8 Rpπq2δ By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels 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 ¡ 0 such that | p q| ¤ A P f x p1 |x|1 q for all x R

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Theorem Neither the Poisson kernel nor the Fejer kernel belong to S(R). For each R ¡ 0 and each y ¡ 0 the Fejer kernel FR and the Poisson kernel Py are continuous functions of moderate decrease with 1. The conjugate Poisson kernel Qy is not a continuous function of moderate decrease. p q y ¡ Proof(For Poisson Kernel) Py x πpx2 y 2q for y 0 If 0 y 1,

y |P pxq| | | y 2pp x q2 q πy y 1 1 p qp x2 q πy 1 y 2

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels If 0 y 1 then

y 2 1 1 ¡ 1 y 2 x2 ¡ x2 y 2 x2 1 ¡ x2 1 y 2 Then 1 |P pxq| y p qp x2 q πy 1 y 2 1 πypx2 1q

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels If y ¥ 1

y ¥ 1 y 2 ¥ 1 x2 y 2gleqx2 1

Then y |P pxq| | | y 2pp x q2 q πy y 1 ¤ y πpx2 1q

Hence, # 1 if 0 y 1, A πy y y ¥ π if y 1

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Kernels

Theorem

Let SR f denote the partial Fourier integral of f, deﬁned by » R 2πixξ SR f pxq fˆpξqe dξ (15) ¡R ³ p q ¦ p q 1 R p q Then SR f x DR f and FR x R 0 Dt x dt.

Proof: »

DR ¦ f DR px ¡ yqf pyqdy (16) »R » R e2πpx¡yqξdξf pyqdy (17) ¡ »R »R » R R e¡2πiyξf pyqdye2πixξdξ f pˆξqe2πixξdξ ¡R R ¡R (18)

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels (19) Kernels

³ p q 1 R p q Also, FR x R 0 Dt x dt because,

» » 1 R 1 R sinp2πtxq D pxqdt dt (20) R t R πx 0 ¢0 § §R 1 ¡1 cosp2πtxq § § (21) R πx 2πx 0 1 pcosp0q ¡ cosp2πRxqq (22) R2π2x2 psinpπRxq2 F pxq (23) Rπ2x2 R

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels 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 R as R Ñ 8. Therefore, the integral Cesaro means of a function of moderate decrease converge to f as R Ñ 8 Theorem » 1 R σR f pxq St f pxqdt FR ¦ f pxq Ñ f pxq (24) R 0 The convergence is both uniform and in Lp

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Integral Cesaro Means

We will prove for p=1.We will show,

lim ||FR ¦ f ¡ f ||1 0 (25) RÑ8

§» § § § § § |FR pxq ¦ f pxq ¡ f pxq| § f px ¡ yqFR pyqdy ¡ f pxq§ (26) §» R § § § § § § pf px ¡ yq ¡ f pxqqFR pyqdy§ (27) » R

¤ |pf px ¡ yq ¡ f pxqq| |FR pyq| dy (28) » » R

||FR ¦ f ¡ f ||1 ¤ |pf px ¡ yq ¡ f pxqq| |FR pyq| dydx (29) R R (30)

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Integral Cesaro Means

By Fubini,

||FR ¦ f ¡ f ||1 ¤ ||pf px ¡ yq ¡ f pxqq| |1 |FR pyq| dy (31) » R

||pf px ¡ yq ¡ f pxqq| |1 |FR pyq| dy (32) »|y|¡δ

||pf px ¡ yq ¡ f pxqq| |1 |FR pyq| dy (33) |y| δ

I1 I2 (34) ³ ||p p ¡ q ¡ p qq| | | p q| Notice, I2 |y| δ f x y f x 1 FR y dy can be made arbitrarily small by continuity of f.

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Integral Cesaro Means

³ ||p p ¡ q ¡ p qq| | | p q| Also, I1 |y|¡δ f x y f x 1 FR y dy can be made arbitrarily small since FR is an approximation to the identity. This concludes the proof, and we have » 1 R σR f pxq St f pxqdt FR ¦ f pxq Ñ f pxq (35) R 0

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels 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 R and fˆpξq 0

» 1 R F ¦ f D pxqdt ¦ f (36) R R t » 0 1 R pD pxq ¦ f qdt (37) R t »0 ¢» 1 R t fˆpξqe2πixξdξ dt (38) R 0 ¡t » £» R R 1 fˆpξqe2πixξ dt dξ (39) ¡R |ξ| R

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Fourier Transform of the Fejer Kernel

» ¢ R |ξ| F ¦ f fˆpξqe2πixξ 1 ¡ dξ (40) R R » ¡R¢ 8 | | ˆ ξ 2πixξ f pξq 1 ¡ χr¡R,Rspξqe dξ (41) ¢¡8 ¢ R |ξ| fˆpξq 1 ¡ χr¡ spξq q (42) R R,R Now we can take the transform, ¢ | | { ˆ ξ FR ¦ f f pξq 1 ¡ χr¡R,Rspξq (43) ¢ R | | { ˆ ˆ ξ FR pξqf pξq f pξq 1 ¡ χr¡R,Rspξq (44) ¢ R { |ξ| F pξq 1 ¡ χr¡ spξq (45) R R R,R

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Connection with Complex Variables

2 Consider a real valued function f P L pRq. Let F pz) be twice the analytic extension of f to the upper half plane 2 R tz x iy : y ¡ 0u Then F pzq is given explicitly by the well known Cauchy Integral Formula: » 1 f ptq F pzq dt (46) πi R t ¡ z If we separate F pzq into its real and imaginary parts, we get » 1 f ptq F pzq dt (47) πi R pt ¡ xq ¡ iy

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Connection with Complex Variables

» 1 f ptqppt ¡ xq iyq F pzq dt (48) p ¡ q2 2 » πi R t x y » p qp ¡ q p q i f t x t 1 f t y 2 2 dt 2 2 dt (49) π R pt ¡ xq y π R pt ¡ xq y f ¦ Py pxq ipf ¦ Qy pxqq (50) u iv (51)

The function u is called the harmonic extension of f to the upper half plane , while v is called the harmonic conjugate of u.

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Theorem 1 2 Qy R L pRq, but Qy P L pRq

Proof » » » 0 ¡ 8 | x | x x 2 2 dx 2 2 dx 2 2 R πpx y q ¡8 πpx y q 0 πpx y q » 2 » y 1 1 8 1 1 ¡ du du 2π u 2 2π u 8» » y 1 8 1 8 1 1 du du 2π 2 u 2 2π u » y y 1 8 1 du π y 2 u 1 ln|u|8 π y 2 8

So Qy pxq R L1pRq By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Connection with Complex Variables

The conjugate Poisson Kernel is not in L1pRq and is not a function of moderate decrease, so we deﬁne its Fourier transform as a principal value: » R ˆ ¡2πiξx Qy pξq lim e Qy pxqdx (52) RÑ8 ¡ » R R x lim e¡2πiξx dx (53) Ñ8 2 2 R ¡R πpx y q (54)

p q ¡2πiξz z Now, consider the function f z e πpz2 y 2q and apply residue theorem for a well chosen contour in the complex plane.

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

e¡2πiξpiyq Respf pzq, iyq (55) 2π e2πiξy for ξ o (56) 2π

e¡2πiξp¡iyq Respf pzq, ¡iyq (57) 2π e¡2πiξy for ξ ¡ 0 (58) 2π

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Connection with complex variables

Therefore

ˆ ¡2πy|ξ| Qy pξq ¡isgnpξqe (59)

Deﬁnition The Hilbert Transform³ H is deﬁned by : p q 1 f pyq P 2p q Hf x π limÑ0 |x¡y|¡ x¡y dy for f L R

Deﬁnition Hfˆ pξq ¡isgnpξqfˆpξq

We can see that the Hilbert transform can be written as the 1 principal value of the convolution of f and πx . It is the response to f of a linear time invariant ﬁlter(Hilbert Transformer) having 1 impulse response πx

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels 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 R 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 ˆ we take the limit of Qy pξq as y goes to zero, we get ˆ Qy pξq Ñ ¡isgnpξq (60)

Then as y Ñ 0, Qy pxq ¦ f approaches the Hilbert transform Hf in 2 L pRq

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels Applications

Consider the initial value problem for the heat equation: ut ¡ uxx 0 in Rxp0, 8q with u g on Rxpt 0q 2 If we take the Fourier transform of u in x we get:u ˆt y uˆ 0 for t ¡ 0 andu ˆ gˆ for t 0 Consequently,u ˆ e¡ty 2 gˆ and ¡ty 2 g¦F ˆ ¡ty 2 u e gˆ therefore u 1 where F e . But then p2πq 2 F pe¡ty 2 q (61) » 1 ixy¡ty 2 1 e dy (62) p2πq 2 R 1 ¡x2 e 4t (63) 2t Therefore, » p q 1 ¡|x¡y|2 p q P ¡ u x, t 1 e g y dy x R, t 0 (64) p4πtq 2 R

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels References

Harmonic Analysis:from Fourier to Wavelets. Cristina Pereyra and Lesly Ward. AMS, 2010. Partial Diﬀerential Equations. Lawrance Evans.AMS, 1998. Fourier Analysis. Stein and Shakarchi.Princetton University Press, 2003.

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels