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Math Help - Fourier Coefficient

  1. #1
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    Fourier Coefficient

    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.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kinkong View Post
    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?
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    yes it is a Hilbert space...i think it is Hilbert basis too...i couldnt even understand the question properly
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    Quote Originally Posted by kinkong View Post
    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.
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    Please can you help me to solve it. i coult get it from your hint. Thanks
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  6. #6
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    Quote Originally Posted by kinkong View Post
    Please can you help me to solve it.
    Bessel's inequality says that \sum|\langle x,e_k\rangle|^2 \leqslant \|x\|^2. The aim now is to prove by contradiction that there are fewer than m^2\|x\|^2 values of k for which |\langle x,e_k\rangle|>1/m.

    So suppose that there are at least m^2\|x\|^2 values of k for which |\langle x,e_k\rangle|>1/m. The the sum \sum|\langle x,e_k\rangle|^2 will be greater than _____ ?
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  7. #7
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    Quote Originally Posted by Opalg View Post
    Bessel's inequality says that \sum|\langle x,e_k\rangle|^2 \leqslant \|x\|^2. The aim now is to prove by contradiction that there are fewer than m^2\|x\|^2 values of k for which |\langle x,e_k\rangle|>1/m.

    So suppose that there are at least m^2\|x\|^2 values of k for which |\langle x,e_k\rangle|>1/m. The the sum \sum|\langle x,e_k\rangle|^2 will be greater than _____ ?
    Will it be greater than ||x||^2??
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