Filtered integration rules for finite Hilbert transforms

TL;DR

This paper introduces a new quadrature rule based on filtered polynomial approximation for efficiently computing finite Hilbert transforms, with proven convergence and numerical validation.

Contribution

It proposes a novel filtered quadrature rule for finite Hilbert transforms, with theoretical convergence analysis and comprehensive numerical comparisons.

Findings

The new rule converges in weighted uniform norm for certain Besov spaces.

Numerical tests show the rule's accuracy and efficiency.

Comparison with existing methods demonstrates its competitive performance.

Abstract

A product quadrature rule, based on the filtered de la Vall\'ee Poussin polynomial approximation, is proposed for evaluating the finite Hilbert transform in [-1; 1]. Convergence results are stated in weighted uniform norm for functions belonging to suitable Besov type subspaces. Several numerical tests are provided, also comparing the rule with other formulas known in literature.