A Riemann Sum Upper Bound in the Riemann--Lebesgue Theorem

Abstract

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