A Riemann sum upper bound in the Riemann-Lebesque theorem

Details A Riemann sum upper bound in the Riemann-Lebesque theorem A Riemann sum upper bound in the Riemann-Lebesque theorem

1997-02-05

arxiv.org/abs/funct-an/9702003v1

Mathematics

Functional Analysis

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

Scientific paper

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.

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