An <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:math> algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero

Rafeh, Rehan,; Bremer, James

doi:10.1016/j.acha.2022.02.001

Cited by 2 publications

(2 citation statements)

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“…In a further work, Bonami and Karoui used the Legendre complete elliptic integral of the first kind to provide explicit approximations of the function and their eigenvalues [23]. Many other works have considered the numerical computation of the PSWFs, both from the integral or differential equation approach [24][25][26][27][28][29][30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Baddour,

Sun

2023

Symmetry

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Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step.

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Section: Introductionmentioning

confidence: 99%

“…Previously in this paper, we implemented the band-limiting operator in the continuous time domain; however, here, we need to interpret it for the discrete-time domain. To implement a band-limiting operator to frequency range [−W, W], then we use Equation (30) with the given bandlimit, so that:…”

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confidence: 99%

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Baddour,

Sun

2023

Symmetry

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On the numerical evaluation of the prolate spheroidal wave functions of order zero

Bremer

2022

Applied and Computational Harmonic Analysis

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An O(1) algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero

Cited by 2 publications

References 12 publications

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

On the numerical evaluation of the prolate spheroidal wave functions of order zero

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