Spectral approximation of convolution operators of Fredholm type

TL;DR

This paper introduces a numerically stable spectral approximation method for Fredholm convolution operators, enabling faster convolutions and spectral analysis of these operators, which was previously difficult.

Contribution

The authors develop a novel spectral approximation algorithm for Fredholm convolution operators that improves computational efficiency and spectral analysis capabilities.

Findings

Faster convolution computations compared to existing methods

Enables spectral analysis of Fredholm convolution operators

Provides a stable and efficient numerical approach

Abstract

We have developed a method for constructing spectral approximations for convolution operators of Fredholm type. The algorithm we propose is numerically stable and takes advantage of the recurrence relations satisfied by the entries of such a matrix approximation. When used for computing the Fredholm convolution of two given functions, such approximations produce the convolution more rapidly than the state-of-the-art methods. The proposed approximation also leads to a spectral method for solving the Fredholm convolution integral equations and enables the computation of eigenvalues and pseudospectra of Fredholm convolution operators, which is otherwise intractable with existing techniques.