Some notes on the trapezoidal rule for Fourier type integrals

TL;DR

This paper analyzes the error of the trapezoidal rule for Fourier integrals using double exponential transformations, leading to the development of an automatic, reliable numerical integration algorithm.

Contribution

It introduces a new error analysis framework for the trapezoidal rule with double exponential transformations and designs an automatic integrator without prior function knowledge.

Findings

The proposed method achieves accurate Fourier integral computations.

Numerical examples confirm the reliability of the automatic integrator.

The approach allows a priori selection of step length and nodes.

Abstract

This paper deals with the error analysis of the trapezoidal rule for the computation of Fourier type integrals, based on two double exponential transformations. The theory allows to construct algorithms in which the steplength and the number of nodes can be a priori selected. The analysis is also used to design an automatic integrator that can be employed without any knowledge of the function involved in the problem. Several numerical examples, which confirm the reliability of this strategy, are reported.