A Riemann sum upper bound in the Riemann-Lebesque theorem

Maurice H. P. M. van Putten

A Riemann sum upper bound in the Riemann-Lebesque theorem

funct-an 2008-02-03 v1 Functional Analysis

Authors: Maurice H. P. M. van Putten

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.

Keywords

multiplicative functions upper bounds bound

Cite

@article{arxiv.funct-an/9702003,
  title  = {A Riemann sum upper bound in the Riemann-Lebesque theorem},
  author = {Maurice H. P. M. van Putten},
  journal= {arXiv preprint arXiv:funct-an/9702003},
  year   = {2008}
}

Comments

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

A Riemann sum upper bound in the Riemann-Lebesque theorem

Abstract

Keywords

Cite

Comments

Related papers