A new friendly method of computing prolate spheroidal wave functions and wavelets

Walter, Gilbert G.; Soleski, Tatiana

doi:10.1016/j.acha.2005.04.001

Cited by 55 publications

(53 citation statements)

References 6 publications

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“…We also notice that for all c > 0, So far, intensive work has been devoted to developing efficient methods for computing the PSWFs and their eigenvalues (see, e.g., [4,29,8,13,16]). A package of Matlab programs for manipulating PSWFs is also available (cf.…”

Section: Lemma 22 For Anymentioning

confidence: 99%

“…Most notably, a series of papers by Slepian et al [25,19,26] and the recent works by Xiao and Rokhlin et al [33,32,24,22] have shown that the PSWFs are a natural and optimal apparatus for approximating bandlimited functions. Recently, there has been a growing interest in developing numerical methods using PSWFs as basis functions, which include in particular the PSWF wavelets [31,30,29] and the PSWF spectral/spectral-element methods [7,8,3,10,18,17,27].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of spectral approximations using prolate spheroidal wave functions

Wang

2009

Math. Comp.

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Abstract. In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of non-periodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectral-Galerkin method is proposed and analyzed for elliptic-type equations. Illustrative numerical results consistent with the theoretical analysis are also presented.

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Section: Lemma 22 For Anymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of spectral approximations using prolate spheroidal wave functions

Wang

2009

Math. Comp.

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“…Many important properties as well as computational issues of PSWFs are made possible by this discovery. Nevertheless, PSWFs have been and still regarded as mysterious functions, see [16]. This is due to the facts that they are difficult to compute and they do not have a standard representation in terms of elementary functions.…”

Section: Introductionmentioning

confidence: 98%

“…Recently, an intensive work has been done in deriving new powerful or efficient methods for computing the eigenvalues and the values of PSWFs, see [3,10,16,[19][20][21]. The methods of [10,16] use the sinc basis functions of c-band limited functions to discretisize the operator (4) and provides approximations of the eigenvalues λ n (c) and the values of PSWFs at the sampling points x N = Nπ/c, N ∈ Z. A first standard and simple representation of PSWFs based on the Shannon sampling theorem has been given in [16].…”

Section: Introductionmentioning

confidence: 99%

New efficient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues

Karoui

Moumni

2008

Applied and Computational Harmonic Analysis

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In this paper, we describe different methods of computing the eigenvalues associated with the prolate spheroidal wave functions (PSWFs). These eigenvalues play an important role in computing the values of PSWFs as well as in the different numerical applications based on these later. The methods given in this work are accurate, fast and valid for small as well as for large values of the bandwidth c of the PSWFs. Moreover, we provide the reader with a method for computing the exact values of PSWFs at the Shannon sampling points. A Shannon sampling theorem with a better decaying sampling basis functions is used to provide a standard representation of the PSWFs. Moreover, we provide the reader with a new fast and accurate method for computing the PSFWFs which is valid over the real line. Based on this method, we develop asymptotic expansions of the PSWFs. Some numerical examples are given to illustrate the results of this paper.

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“…Fauvet et al [9] also studied an eigenvalue problem for a special non-Fuchsian secondorder differential equation called the prolate spheroidal wave equation [27] on a finite interval, using the differential Galois theory. They analyzed the Stokes phenomenon and clarified a relation between solutions of the eigenvalue problem and the differential Galois group.…”

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confidence: 99%

Galoisian approach for a Sturm-Liouville problem on the infinite interval

Blázquez-Sanz

Yagasaki

2012

Methods and Applications of Analysis

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Abstract. We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval (−∞, ∞). Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with three examples for a stationary Schrödinger equation having a generalized Hulthén potential; a linear stability equation for a traveling front in the Allen-Cahn equation; and an eigenvalue problem related to the Lamé equation.

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A new friendly method of computing prolate spheroidal wave functions and wavelets

Cited by 55 publications

References 6 publications

Analysis of spectral approximations using prolate spheroidal wave functions

Analysis of spectral approximations using prolate spheroidal wave functions

New efficient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues

Galoisian approach for a Sturm-Liouville problem on the infinite interval

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