On the Convergence of Numerical Integration as a Finite Matrix Approximation to Multiplication Operator

TL;DR

This paper investigates the convergence of numerical integration methods approximating multiplication operators with finite matrices, extending existing results to unbounded functions and using operator theory for analysis.

Contribution

It extends convergence analysis of matrix-based numerical integration to unbounded functions and domains using strong resolvent convergence and operator inequalities.

Findings

Convergence established for unbounded functions and domains.

Application of strong resolvent convergence theory.

Use of Jensen's operator inequality for improper integrals.

Abstract

We study the convergence of a family of numerical integration methods where the numerical integral is formulated as a finite matrix approximation to a multiplication operator. For bounded functions, the convergence has already been established using the theory of strong operator convergence. In this article, we consider unbounded functions and domains which pose several difficulties compared to the bounded case. A natural choice of method for this study is the theory of strong resolvent convergence which has previously been mostly applied to study the convergence of approximations of differential operators. The existing theory already includes convergence theorems that can be used as proofs as such for a limited class of functions and extended for wider class of functions in terms of function growth or discontinuity. The extended results apply to all self-adjoint operators, not just multiplication operators. We also show how Jensen's operator inequality can be used to analyse the convergence of an improper numerical integral of a function bounded by an operator convex function.