On the Equivalence of Fourier Expansion and Poisson Summation Formula for the Series Approximation of the Exponential Function

TL;DR

This paper demonstrates that Fourier expansion and Poisson summation are fundamentally equivalent methods for approximating the exponential function, simplifying the understanding of their relationship in series approximation techniques.

Contribution

It establishes the theoretical equivalence between Fourier expansion and Poisson summation for the exponential function, clarifying their relationship in series approximation.

Findings

Poisson summation reduces to Fourier expansion in this context

The equivalence simplifies the analysis of exponential series approximations

Provides a unified view of two fundamental mathematical techniques

Abstract

In this short note we show the equivalence of Fourier expansion and Poisson summation approaches for the series approximation of the exponential function $\exp ({-{t^2}/4})$. The application of the Poisson summation formula is shown to reduce to that of the Fourier expansion method.