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