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Math Help - Minimum Property of Fourier Coefficients

  1. #1
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    Minimum Property of Fourier Coefficients

    Dear Colleagues,

    Could you please help me in solving the problem in the pdf attachment.
    Because of LATEX technical problems I attached a file containing the problem.


    Best Regards.
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  2. #2
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    If y = \textstyle\sum\beta_je_j then \textstyle\|x-y\|^2 = \bigl\langle x-\sum\beta_je_j, x-\textstyle\sum\beta_je_j\bigr\rangle, so that

    \textstyle\|x-y\|^2 = \|x\|^2 - 2\text{Re}\sum\bigl(\langle x,e_j\rangle\beta_je_j\bigr)\overline{\beta_j} + \sum|\beta_je_j|^2.

    Also, \textstyle\sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 = \sum|\langle x,e_j\rangle|^2 - 2\text{Re}\sum\bigl(\langle x,e_j\rangle\beta_je_j\bigr)\overline{\beta_j} + \sum|\beta_je_j|^2.

    Subtract that equation from the previous one to see that \|x-y\|^2 is smallest when \textstyle\sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 = 0.
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  3. #3
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    Thank you for your reply. But I proved that -as I mentioned in the attachments- and I need your help in proving the converse.


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  4. #4
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    Quote Originally Posted by raed View Post
    Thank you for your reply. But I proved that -as I mentioned in the attachments- and I need your help in proving the converse.
    Surely the equation \textstyle\|x-y\|^2 = \|x\|^2 - \sum|\langle x,e_j\rangle|^2 + \sum\bigl|\langle x,e_j\rangle - \beta_j\bigr|^2 proves both the result and the converse?
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