A generalized spectral concentration problem and the varying masks algorithm

TL;DR

This paper extends the spectral concentration problem to arbitrary masks, introduces a robust algorithm for its solution, and demonstrates its effectiveness across various domains and shapes.

Contribution

It generalizes the spectral concentration problem with new theoretical results and proposes a versatile, efficient algorithm for practical implementation.

Findings

The generalized problem is well-posed with arbitrary masks.

Standard eigen-algorithms face eigenvalue clustering issues.

The new algorithm is robust, efficient, and applicable in any dimension.

Abstract

In this paper we generalize the spectral concentration problem as formulated by Slepian, Pollak and Landau in the 1960s. We show that a generalized version with arbitrary space and Fourier masks is well-posed, and we prove some new results concerning general quadratic domains and gaussian filters. We also propose a more general splitting representation of the spectral concentration operator allowing to construct quasi-modes in some situations. We then study its discretization and we illustrate the fact that standard eigen-algorithms are not robust because of a clustering of eigenvalues. We propose a new alternative algorithm that can be implemented in any dimension and for any domain shape, and that gives very efficient results in practice.