Fourier Coefficient

**kinkong** · May 26th 2011, 09:41 PM

Please can you guys help me to solve the following question. I couldnt even get the question.
Q. Show that an element x of an inner product space X cannot have "too many" Fourier coefficeints <x,e_k> which are "big"; here, (e_k) is given orthonormal sequence; more precisely, show that number n_m of <x,e_k> such that |<x,e_k>| > 1/m must satisfy n_m < m^2 ||x||^2.

**Drexel28** · May 26th 2011, 10:27 PM

Originally Posted by kinkong

Please can you guys help me to solve the following question. I couldnt even get the question.
Q. Show that an element x of an inner product space X cannot have "too many" Fourier coefficeints <x,e_k> which are "big"; here, (e_k) is given orthonormal sequence; more precisely, show that number n_m of <x,e_k> such that |<x,e_k>| > 1/m must satisfy n_m < m^2 ||x||^2.

Is this a Hilbert space? Is that orthonormal basis a Hamel or Schauder basis?

**kinkong** · May 26th 2011, 11:36 PM

yes it is a Hilbert space...i think it is Hilbert basis too...i couldnt even understand the question properly

**Opalg** · May 27th 2011, 12:08 AM

Originally Posted by kinkong

Please can you guys help me to solve the following question. I couldnt even get the question.
Q. Show that an element x of an inner product space X cannot have "too many" Fourier coefficeints <x,e_k> which are "big"; here, (e_k) is given orthonormal sequence; more precisely, show that number n_m of <x,e_k> such that |<x,e_k>| > 1/m must satisfy n_m < m^2 ||x||^2.

Hint: Bessel's inequality.