Spatiospectral concentration in the Cartesian plane

TL;DR

This paper extends Slepian's time-frequency concentration problem to two-dimensional Cartesian space, providing a framework for constructing functions that are optimally localized in both space and frequency for scientific applications.

Contribution

It introduces a method to find orthogonal functions with optimal spatiospectral concentration in 2D Cartesian domains, generalizing previous 1D and spherical cases.

Findings

Derived a Fredholm integral equation for the functions

Identified special algorithms for circularly symmetric domains

Applicable to geophysics and astronomy signal analysis

Abstract

We pose and solve the analogue of Slepian's time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.