Spectral approximation of convolution operator

TL;DR

This paper introduces a unified, numerically stable framework for approximating Volterra convolution operators using orthogonal polynomial-based matrices, enhancing computational efficiency and stability.

Contribution

It presents a novel, unified approach for constructing stable matrix approximations of convolution operators using classical orthogonal polynomials, including Laguerre polynomials.

Findings

Develops algorithms exploiting recurrence relations and symmetry.

Provides stable matrix approximation techniques for Volterra convolution.

Includes discussion on Laguerre-based convolution matrices.

Abstract

We develop a unified framework for constructing matrix approximations to the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on $[-1, 1]$. The numerically stable algorithms we propose exploit recurrence relations and symmetric properties satisfied by the entries of these convolution matrices. Laguerre-based convolution matrices that approximate Volterra convolution operator defined by functions on $[0, \infty]$ are also discussed for the sake of completeness.