Quadrature formulae for the positive real axis in the setting of Mellin analysis: Sharp error estimates in terms of the Mellin distance

TL;DR

This paper develops sharp error estimates for Mellin quadrature formulas on the positive real axis, relating convergence rates to the function's Mellin distance and characterizing these rates in Mellin-Sobolev and Mellin-Hardy spaces.

Contribution

It provides optimal order error bounds for Mellin quadrature formulas and characterizes convergence rates via Mellin-based function spaces.

Findings

Error estimates are of the best possible order.

Convergence rates are characterized in Mellin-Sobolev and Mellin-Hardy spaces.

Numerical experiments confirm theoretical results.

Abstract

The general Poisson summation formula of Mellin analysis can be considered as a quadrature formula for the positive real axis with remainder. For Mellin bandlimited functions it becomes an exact quadrature formula. Our main aim is to study the speed of convergence to zero of the remainder for a function $f$ in terms of its distance from a space of Mellin bandlimited functions. The resulting estimates turn out to be of best possible order. Moreover, we characterize certain rates of convergence in terms of Mellin--Sobolev and Mellin--Hardy type spaces that contain $f$. Some numerical experiments illustrate and confirm these results.