M\"obius-transformed trapezoidal rule for polynomial weights

TL;DR

This paper introduces a M"obius-transformed trapezoidal rule for numerical integration that achieves optimal convergence rates for polynomially weighted integrals, verified through theoretical analysis and numerical experiments.

Contribution

It presents a novel quadrature method combining M"obius transformation with the trapezoidal rule, achieving optimal convergence for weighted integrals with minimal evaluation requirements.

Findings

Achieves optimal convergence rate for polynomially weighted integrals.

Method requires only pointwise evaluations at fixed nodes.

Numerical experiments confirm theoretical convergence rates.

Abstract

This work studies numerical integration by the M\"obius-transformed trapezoidal rule, which combines the classical trapezoidal rule with a change of variables induced by a M\"obius transformation that maps the unit circle onto the real line. It is shown that this method achieves the optimal convergence rate for a polynomially weighted integral over the real line if the integrand lives in a related polynomially weighted Sobolev space with positive integer smoothness index. This result can also be generalized in a slightly weaker form for fractional smoothness indices via complex interpolation of function spaces. The algorithm only requires pointwise evaluations of the weight and the target integrand at prescribed nodes that do not depend on the integrand and weight in question. The established theoretical convergence rates are verified by numerical experiments.