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Thread: Fourier Coefficients

  1. #1
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    Fourier Coefficients

    Dear Colleagues,

    Could you please help me in solving the following problem:
    Show that an element $\displaystyle x$ of an inner product space $\displaystyle X$ cannot have "too many" Fourier coefficients $\displaystyle <x, e_{k}>$ which are "big"; here, $\displaystyle (e_{k})$ is a given orthonormal sequence; more precisely, show that the number $\displaystyle n_{m}$ of $\displaystyle <x, e_{k}>$ such that $\displaystyle |<x, e_{k}>|>1/m$ must satisfy $\displaystyle n_{m}<m^{2}||x||^{2}$.

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


    Best Regards.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Use Parsevals' Identity.
    Parseval's identity - Wikipedia, the free encyclopedia

    we know that the sum

    $\displaystyle \sum_{n=1}^{\infty}<x,e_n>$

    Is finite.

    Since the series converges we know that the limit

    $\displaystyle \lim_{n \to \infty}<x,e_n>=0$
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  3. #3
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    Thank you for your reply. In fact, Parseval's identity does not hold here since $\displaystyle (e_{k})$ is an orthonormal sequence not basis in the inner product space $\displaystyle X$.

    Regards.
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Sorry You need Bessel's inequality then

    Bessel's inequality - Wikipedia, the free encyclopedia
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