A common framework for some techniques in Applied mathematics

TL;DR

This paper presents a unified framework for four fundamental applied mathematics techniques—Fourier series, Fourier transform, Laplace transform, and Fourier-Laplace transform—using eigenvalue problems of differential operators.

Contribution

It introduces a common eigenvalue problem-based approach to derive and understand these techniques, linking them through spectral theory.

Findings

Unified eigenvalue problem framework for Fourier and Laplace techniques

Derivation of transforms from differential operator spectra

Enhanced understanding of technique interrelations

Abstract

The objective in this paper is to demonstrate that four of the most used techniques in applied mathematics, viz., Fourier series, Fourier transform, Laplace transform and the Fourier-Laplace transform can be introduced using eigenvalue problems for first order differential operators with discrete/continuous spectra.