[Paper Review] On generalized prolate spheroidal functions

Philip Greengard, Kirill Serkh|arXiv (Cornell University)|Nov 7, 2018

Image and Signal Denoising Methods18 references4 citations

TL;DR

This paper develops analytical and numerical tools for generalized prolate spheroidal functions (GPSFs) in higher dimensions, extending the one-dimensional framework of Slepian et al. It enables efficient eigenvalue computation, quadrature construction, and interpolation, significantly improving the practical usability of GPSFs for bandlimited functions on multidimensional intervals.

ABSTRACT

Prolate spheroidal wave functions provide a natural and effective tool for computing with bandlimited functions defined on an interval. As demonstrated by Slepian et al., the so called generalized prolate spheroidal functions (GPSFs) extend this apparatus to higher dimensions. While the analytical and numerical apparatus in one dimension is fairly complete, the situation in higher dimensions is less satisfactory. This report attempts to improve the situation by providing analytical and numerical tools for GPSFs, including the efficient evaluation of eigenvalues, the construction of quadratures, interpolation formulae, etc. Our results are illustrated with several numerical examples.

Motivation & Objective

• To address the incomplete analytical and numerical framework for generalized prolate spheroidal functions (GPSFs) in higher dimensions.
• To extend the one-dimensional prolate spheroidal function theory—effective for bandlimited functions on intervals—to multidimensional domains.
• To provide practical computational tools such as eigenvalue evaluation, quadrature rules, and interpolation formulae for GPSFs.
• To improve the numerical efficiency and accuracy of working with bandlimited functions in higher-dimensional intervals.

Proposed method

• Develops analytical expressions and numerical algorithms for computing GPSF eigenvalues in higher dimensions.
• Constructs quadrature rules based on GPSF eigenfunctions to enable accurate integration of bandlimited functions.
• Derives interpolation formulae using GPSF bases to reconstruct bandlimited functions from sampled data.
• Employs spectral methods and special function theory to ensure stability and convergence in multidimensional settings.
• Validates the approach through numerical examples demonstrating accuracy and efficiency in multidimensional function approximation.

Experimental results

Research questions

• RQ1 How can GPSF eigenvalues be efficiently computed in higher-dimensional domains?
• RQ2 What quadrature rules based on GPSFs ensure high-accuracy integration of bandlimited functions in multidimensions?
• RQ3 How can GPSF-based interpolation formulae be constructed to reconstruct bandlimited functions from discrete samples?
• RQ4 What are the convergence and stability properties of GPSF-based numerical methods in higher dimensions?
• RQ5 How do the proposed tools compare in performance and accuracy to existing methods for bandlimited functions?

Key findings

• The paper successfully extends the one-dimensional prolate spheroidal function framework to higher dimensions with analytically and numerically tractable tools.
• Efficient algorithms for GPSF eigenvalue computation are developed, enabling practical implementation in multidimensional settings.
• Quadrature rules based on GPSFs achieve high accuracy for integrating bandlimited functions over multidimensional intervals.
• Interpolation formulae using GPSF bases demonstrate high convergence rates and stability in numerical experiments.
• Numerical examples confirm the effectiveness and robustness of the proposed tools across various multidimensional test cases.