A Riemann sum upper bound in the Riemann-Lebesque theorem

TL;DR

This paper presents a novel proof of an upper bound for Fourier coefficients in the Riemann-Lebesque theorem using Riemann sums, providing an alternative approach to the classical Lebesgue-based proof.

Contribution

It introduces a Riemann sum-based upper bound for Fourier coefficients, offering a different perspective from traditional Lebesgue integration proofs.

Findings

Established the inequality $2\pi|c_k(f)| ext{ } extless ext{ } S_k(f)-s_k(f)$ for Fourier coefficients.

Provided a proof suitable for educational purposes in applied mathematics courses.

Demonstrated the applicability of Riemann sums in Fourier analysis contexts.

Abstract

The Riemann-Lebesque Theorem is commonly proved in a few strokes using the theory of Lebesque integration. Here, the upper bound $2\pi|c_k(f)|\le S_k(f)-s_k(f)$ for the Fourier coefficients $c_k$ is proved in terms of majoring and minoring Riemann sums $S_k(f)$ and $s_f(k)$, respectively, for Riemann integrable functions $f(x)$. This proof has been used in a course on methods of applied mathematics.