Sampling theory approach to prolate spheroidal wavefunctions

Khare, Kedar; George, Nicholas

doi:10.1088/0305-4470/36/39/303

Cited by 53 publications

(56 citation statements)

References 18 publications

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“…While the Bell Labs theory addressed several aspects of time-frequency localization, it did not address potentially useful connections between time-frequency localization and sampling. Examples of such connections have been established only relatively recently, independently by Khare and George [21] and by Shen and Walter [41]. Specifically, both sets of authors showed that the integer samples of the eigenfunctions of P Q T P are eigenvectors of the matrix A k = T /2 −T /2 sinc (t − k) sinc (t − ) dt.…”

mentioning

confidence: 99%

Time-Frequency Localization and Sampling of Multiband Signals

Izu

Lakey

2009

Acta Appl Math

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This paper begins with a review of some classical work of Landau, Slepian, Pollak and Widom concerning essentially time-and bandlimited signals and ends reviewing some recent work of Candès, Romberg and Tao that places specific but probabilistic limitations on essential time-and bandlimiting for finite signals and their discrete Fourier transforms. In between we outline some conceptual bridges from the continuous, singleband setting to the finite, multiband setting and pose a number of open problems whose solutions would solidify the connections outlined. These connections involve time-frequency localization of multiband signals and sampling theory for such signals.

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confidence: 99%

Time-Frequency Localization and Sampling of Multiband Signals

Izu

Lakey

2009

Acta Appl Math

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“…Unfortunately, the discretization of (7) by the sinc functions leads to a matrix with low decay coefficients. Consequently, the computation of the eigenvalues and the values of PSWFs at the sampling points by this method is no longer friendly if high precision is required [10]. In this section, we show that by using different values of the Fourier series coefficients of the Legendre polynomials, it is possible to compute fastly and accurately the different values ψ n,c ( Nπ c ), N ∈ Z.…”

Section: Classical Legendre-shannon Methodsmentioning

confidence: 99%

“…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%

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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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“…Most of the material here is collected from different sources [2,3,5,6,11,12,16]; see also [1,4,15,17].…”

Section: Prolate Spheroidal Wave Functionsmentioning

confidence: 99%

A generalization of the prolate spheroidal wave functions

Zayed

2007

Proc. Amer. Math. Soc.

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Abstract. Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of L 2 (−∞, ∞) and L 2 (0, ∞), and the Jacobi polynomials which are an orthogonal basis of a weighted L 2 (−1, 1). The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of L 2 (−1, 1).The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both L 2 (−1, 1) and a subspace of L 2 (−∞, ∞), known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property.The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.

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Sampling theory approach to prolate spheroidal wavefunctions

Cited by 53 publications

References 18 publications

Time-Frequency Localization and Sampling of Multiband Signals

Time-Frequency Localization and Sampling of Multiband Signals

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

A generalization of the prolate spheroidal wave functions

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