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Abstract
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Generalized prolate spheroidal functions (GPSFs) arise naturally in the study of
bandlimited functions as the eigenfunctions of a certain truncated Fourier transform.
In one dimension, the theory of GPSFs (typically referred to as prolate spheroidal
wave functions) has a long history and is fairly complete. Furthermore, more recent
work has led to the development of numerical algorithms for their computation and
use in applications. In this paper we consider the more general problem, extending
the one-dimensional analysis and algorithms to the case of arbitrary dimension.
Specifically, we introduce algorithms for efficient evaluation of GPSFs and their
corresponding eigenvalues, quadrature rules for bandlimited functions, formulae for
approximation via GPSF expansion, and various analytical properties of GPSFs.
We illustrate the numerical and analytical results with several numerical
examples.
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Keywords
generalized prolate spheroidal functions, bandlimited
function approximation, special functions
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Mathematical Subject Classification
Primary: 65D15, 65D20, 65D32
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Milestones
Received: 14 October 2023
Revised: 6 May 2024
Accepted: 11 June 2024
Published: 1 October 2024
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