A Riemann sum upper bound in the Riemann-Lebesque theorem

1997/02/05 by Maurice H. P. M. van Putten, van Putten, Maurice H. P. M.
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Limits and Structures in Graph Theory #funct-an #math.FA

paper · pdf · doi:10.48550/arxiv.funct-an/9702003

LaTex, 2 pages, to appear in the Classroom Notes section in SIAM Review

arxiv created 1997/02/05 · openalex publication_date 1997/02/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Riemann-Lebesque Theorem is commonly proved in a few strokes using the theory of Lebesque integration. Here, the upper bound 2π|c_k(f)|≤ 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.

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