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Generalized prolate spheroidal functions: algorithms and analysis

Philip Greengard

TL;DR

Algorithms for efficient evaluation of GPSFs and their corresponding eigenvalues, quadrature rules for bandlimited functions, formulae for interpolation via GPSF expansion, and various analytical properties of GPSF are introduced.

Abstract

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 interpolation via GPSF expansion, and various analytical properties of GPSFs. We illustrate the numerical and analytical results with several numerical examples.

Generalized prolate spheroidal functions: algorithms and analysis

TL;DR

Algorithms for efficient evaluation of GPSFs and their corresponding eigenvalues, quadrature rules for bandlimited functions, formulae for interpolation via GPSF expansion, and various analytical properties of GPSF are introduced.

Abstract

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 interpolation via GPSF expansion, and various analytical properties of GPSFs. We illustrate the numerical and analytical results with several numerical examples.

Paper Structure

This paper contains 29 sections, 44 theorems, 229 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Mathematical and numerical preliminaries
  3. Spherical harmonics in $\mathbbm{R}^{p+2}$
  4. Jacobi polynomials
  5. Zernike polynomials
  6. Numerical evaluation of Zernike polynomials
  7. Modified Zernike polynomials
  8. Prüfer transform
  9. Generalized prolate spheroidal functions
  10. Basic facts
  11. Eigenfunctions and eigenvalues of $F_c$
  12. The dual nature of GPSFs
  13. Zernike polynomials and GPSFs
  14. Analytical apparatus
  15. Properties of GPSFs
  16. ...and 14 more sections

Key Result

Lemma 2.1

For any real number $a>0$ and for any integer $n>ae$, where $\Gamma(n)$ is defined in (1780).

Figures (9)

  • Figure 1: Eigenvalues of $F_c$ (see (\ref{['4.10']})) for $c=100$ and $p=0$
  • Figure 2: Eigenvalues of $F_c$ (see (\ref{['4.10']})) for $c=50$ and $p=1$
  • Figure 3: Eigenvalues of $F_c$ (see (\ref{['4.10']})) for $c=200$ and $p=2$
  • Figure 4: Plots of GPSFs $\Phi_{0,n}$ (see (\ref{['5.100']})) with $c=50$ and $p=1$
  • Figure 5: Plots of GPSFs $\Phi_{0,n}$ (see (\ref{['5.100']})) with $c=100$ and $p=0$
  • ...and 4 more figures

Theorems & Definitions (68)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.1
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Remark 2.1
  • ...and 58 more