##### Document Text Contents

Page 1

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

CIMPA School on Real and Complex Analysis with Applications,

Buea Cameroun, 1–14 May 2011.

Prolate Spheroidal Wave functions and

Applications.

Abderrazek Karoui in collaboration with Aline Bonami

University of Carthage, Department of Mathematics,

Faculty of Sciences of Bizerte, Tunisia.

March 29, 2011

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 2

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

Outline

1 PSWFs and Properties

PSWFs and PDE

differential and integral operators associated with PSWFs

Some Properties of the PSWFs

2 Computation of the PSWFs

3 Uniform estimates of the PSWFs and their derivatives

WKB method for the PSWFs

Uniform bounds of the PSWFs and their derivatives

Exponential decay of the eigenvalues associated with the

PSWFs

4 Applications of the PSWFs

PSWFs based spectral approximation in Sobolev spaces.

Signal processing applications

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 27

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

WKB method for the PSWFs

Uniform bounds of the PSWFs and their derivatives

Exponential decay of the eigenvalues associated with the PSWFs

Proposition

For any integers n, k ≥ 0, satisfying k(k + 1) ≤ χn, we have∣∣∣ψ(k)n,c (0)∣∣∣ ≤ (√χn)k |ψn,c(0)| , (22)

for n even and k even, and∣∣∣ψ(k)n,c (0)∣∣∣ ≤ (√χn)k−1 ∣∣ψ′n,c(0)∣∣ , (23)

for n odd and k odd. In particular, under the assumption that

q =

c2

χn

< 1, there exists a constant C , depending only on q and

such that for any positive integer k satisfying k(k + 1) ≤ χn, we

have ∣∣∣ψ(k)n,c (0)∣∣∣ ≤ C (√χn)k . (24)

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 28

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

WKB method for the PSWFs

Uniform bounds of the PSWFs and their derivatives

Exponential decay of the eigenvalues associated with the PSWFs

We start from the well-known equality, [Slepian 1964, Rokhlin et

al. (2007)], that for any positive integer n, we have

λn(c) = λ

′ × λ′′, with

λ′ : =

c2n+1(n!)4

2((2n)!)2(Γ(n + 3/2))2

(25)

λ′′ : = exp

(

2

∫ c

0

(ψn,τ (1))

2 − (n + 1/2)

τ

dτ

)

. (26)

Proposition

Let α < 1. There exists a constant Mα with the following property.

For all n and c ≥ 0 such that qn(c) ≤ α, then

sup

x∈[−1,1]

∣∣ψn,c(x)− Pn(x)∣∣ ≤ Mα c2√

n + 1/2

, (27)

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 53

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

PSWFs based spectral approximation in Sobolev spaces.

Signal Processing Applications of the PSWFs

[5] D. L. Donoho and P. B. Stark, Uncertainty principles and signal

recovery, SIAM Journal of Applied mathematics, 49, (1989),

pp.906-931.

[6] Li-Lian Wang, Analysis of Spectral Approximations using

Prolate Spheroidal Wave Functions, Mathematics of Computation

79 (2010), 807-827.

[7] D. Slepian, H. O. Pollak, Prolate spheroidal wave functions,

Fourier analysis, and uncertainty-I, Bell Syst. Tech. J. 40 (1961),

43–63.

[8] D. Slepian, Prolate spheroidal wave functions, Fourier analysis

and uncertainty–IV: Extensions to many dimensions; generalized

prolate spheroidal functions, Bell System Tech. J. 43 (1964),

3009–3057.

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 54

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

PSWFs based spectral approximation in Sobolev spaces.

Signal Processing Applications of the PSWFs

[9] H. J. Landau and H. Widom, Eigenvalue distribution of time

and frequency limiting, J. Math. Anal.Appl., 77, (1980), 469–481.

[10] V. Rokhlin and H. Xiao, Approximate formulae for certain

prolate spheroidal wave functions valid for large values of both

order and band-limit, Appl. Comput. Harmon. Anal. 22, (2007),

105–123.

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

CIMPA School on Real and Complex Analysis with Applications,

Buea Cameroun, 1–14 May 2011.

Prolate Spheroidal Wave functions and

Applications.

Abderrazek Karoui in collaboration with Aline Bonami

University of Carthage, Department of Mathematics,

Faculty of Sciences of Bizerte, Tunisia.

March 29, 2011

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 2

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

Outline

1 PSWFs and Properties

PSWFs and PDE

differential and integral operators associated with PSWFs

Some Properties of the PSWFs

2 Computation of the PSWFs

3 Uniform estimates of the PSWFs and their derivatives

WKB method for the PSWFs

Uniform bounds of the PSWFs and their derivatives

Exponential decay of the eigenvalues associated with the

PSWFs

4 Applications of the PSWFs

PSWFs based spectral approximation in Sobolev spaces.

Signal processing applications

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 27

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

WKB method for the PSWFs

Uniform bounds of the PSWFs and their derivatives

Exponential decay of the eigenvalues associated with the PSWFs

Proposition

For any integers n, k ≥ 0, satisfying k(k + 1) ≤ χn, we have∣∣∣ψ(k)n,c (0)∣∣∣ ≤ (√χn)k |ψn,c(0)| , (22)

for n even and k even, and∣∣∣ψ(k)n,c (0)∣∣∣ ≤ (√χn)k−1 ∣∣ψ′n,c(0)∣∣ , (23)

for n odd and k odd. In particular, under the assumption that

q =

c2

χn

< 1, there exists a constant C , depending only on q and

such that for any positive integer k satisfying k(k + 1) ≤ χn, we

have ∣∣∣ψ(k)n,c (0)∣∣∣ ≤ C (√χn)k . (24)

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 28

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

WKB method for the PSWFs

Uniform bounds of the PSWFs and their derivatives

Exponential decay of the eigenvalues associated with the PSWFs

We start from the well-known equality, [Slepian 1964, Rokhlin et

al. (2007)], that for any positive integer n, we have

λn(c) = λ

′ × λ′′, with

λ′ : =

c2n+1(n!)4

2((2n)!)2(Γ(n + 3/2))2

(25)

λ′′ : = exp

(

2

∫ c

0

(ψn,τ (1))

2 − (n + 1/2)

τ

dτ

)

. (26)

Proposition

Let α < 1. There exists a constant Mα with the following property.

For all n and c ≥ 0 such that qn(c) ≤ α, then

sup

x∈[−1,1]

∣∣ψn,c(x)− Pn(x)∣∣ ≤ Mα c2√

n + 1/2

, (27)

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 53

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

PSWFs based spectral approximation in Sobolev spaces.

Signal Processing Applications of the PSWFs

[5] D. L. Donoho and P. B. Stark, Uncertainty principles and signal

recovery, SIAM Journal of Applied mathematics, 49, (1989),

pp.906-931.

[6] Li-Lian Wang, Analysis of Spectral Approximations using

Prolate Spheroidal Wave Functions, Mathematics of Computation

79 (2010), 807-827.

[7] D. Slepian, H. O. Pollak, Prolate spheroidal wave functions,

Fourier analysis, and uncertainty-I, Bell Syst. Tech. J. 40 (1961),

43–63.

[8] D. Slepian, Prolate spheroidal wave functions, Fourier analysis

and uncertainty–IV: Extensions to many dimensions; generalized

prolate spheroidal functions, Bell System Tech. J. 43 (1964),

3009–3057.

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.

Page 54

Prolate Spheroidal Wave Functions and their Properties

Computation of the PSWFs by Flammer’s method

Uniform estimates of the PSWFs and their derivatives

Applications of the PSWFs

PSWFs based spectral approximation in Sobolev spaces.

Signal Processing Applications of the PSWFs

[9] H. J. Landau and H. Widom, Eigenvalue distribution of time

and frequency limiting, J. Math. Anal.Appl., 77, (1980), 469–481.

[10] V. Rokhlin and H. Xiao, Approximate formulae for certain

prolate spheroidal wave functions valid for large values of both

order and band-limit, Appl. Comput. Harmon. Anal. 22, (2007),

105–123.

Abderrazek Karoui in collaboration with Aline Bonami Prolate Spheroidal Wave functions and Applications.