# Fourier Coefficients

• May 10th 2011, 07:08 AM
raed
Fourier Coefficients
Dear Colleagues,

Show that an element $x$ of an inner product space $X$ cannot have "too many" Fourier coefficients $$ which are "big"; here, $(e_{k})$ is a given orthonormal sequence; more precisely, show that the number $n_{m}$ of $$ such that $||>1/m$ must satisfy $n_{m}.

In fact the most important part -in my opinion- is how we can show that $n_{m}$ is finite?

Best Regards.
• May 10th 2011, 07:19 AM
TheEmptySet
Use Parsevals' Identity.
Parseval's identity - Wikipedia, the free encyclopedia

we know that the sum

$\sum_{n=1}^{\infty}$

Is finite.

Since the series converges we know that the limit

$\lim_{n \to \infty}=0$
• May 10th 2011, 07:28 AM
raed
Thank you for your reply. In fact, Parseval's identity does not hold here since $(e_{k})$ is an orthonormal sequence not basis in the inner product space $X$.

Regards.
• May 10th 2011, 07:34 AM
TheEmptySet
Sorry You need Bessel's inequality then

Bessel's inequality - Wikipedia, the free encyclopedia