Errors and ambiguity in transition from Fourier series to Fourier integrals

TL;DR

This paper investigates the errors and ambiguities involved when converting Fourier series to Fourier integrals, providing estimates for errors and conditions to avoid ambiguities in the transition.

Contribution

It offers a detailed analysis of the errors and ambiguities in transitioning from Fourier series to Fourier integrals, including error estimates and conditions to prevent ambiguity.

Findings

Error estimates for substitution in Fourier series to integrals

Conditions under which ambiguity does not occur

Analysis of the transition's impact on function representation

Abstract

Transition from Fourier series to Fourier integrals is considered and error introduced by ordinary substitution of integration for summing is estimated. Ambiguity caused by transition from discrete function to continuous one is examined and conditions under which this ambiguity does not arise are suggested.