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TitleProlate Spheroidal Wave functions and Applications.
LanguageEnglish
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Table of Contents
                            PSWFs and Properties
	PSWFs and PDE
	differential and integral operators associated with PSWFs
	Some Properties of the PSWFs
Computation of the PSWFs
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
Applications of the PSWFs
	PSWFs based spectral approximation in Sobolev spaces.
	Signal processing applications
                        
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



)
. (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.

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