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Math Help - Proving Parseval's Theorem

  1. #1
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    Proving Parseval's Theorem

    Problem: A) Let \{v_1, v_2, ..., v_3\} be an orthonormal basis for V. For any x,y \epsilon V prove that
    <x,y>= \sum\limits_{i=1}^n <x,v_i> \overline{<y,v_i>}
    B) use (A) to prove that if \beta is an orthonormal basis for V with inner product <*, *> then for any  x,y\epsilon V <\phi_\beta (x),\phi_\beta (y)>' = <[x]_\beta.[y]_\beta>'=<x,y> where <*, *> is the standard product on F^n

    Thoughts:
    I worked out (A) but I am almost certain I did it wrong, however, this is what I did:
    <x,y>= \sum\limits_{i=1}^n <x,v_i> \overline{<y,v_i>} = \sum\limits_{i=1}^n <x,v_i> <v_i,y>
     = \sum\limits_{i=1}^n\sum\limits_{j=1}^n x_j y_{ij} v_{ij} y_j = \sum\limits_{i=1}^n\sum\limits_{j=1}^n x_j y_j =<x,y>
    And I haven't the slightest idea what (B) means, let alone how to prove it using A which I almost certainly did wrong.
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  2. #2
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    Quote Originally Posted by kaelbu View Post
    Problem: A) Let \{v_1, v_2, ..., v_3\} be an orthonormal basis for V. For any x,y \epsilon V prove that
    <x,y>= \sum\limits_{i=1}^n <x,v_i> \overline{<y,v_i>}
    B) use (A) to prove that if \beta is an orthonormal basis for V with inner product <*, *> then for any  x,y\epsilon V <\phi_\beta (x),\phi_\beta (y)>' = <[x]_\beta.[y]_\beta>'=<x,y> where <*, *> is the standard product on F^n

    Thoughts:
    I worked out (A) but I am almost certain I did it wrong, however, this is what I did:
    <x,y>= \sum\limits_{i=1}^n <x,v_i> \overline{<y,v_i>} = \sum\limits_{i=1}^n <x,v_i> <v_i,y>
     = \sum\limits_{i=1}^n\sum\limits_{j=1}^n x_j y_{ij} v_{ij} y_j = \sum\limits_{i=1}^n\sum\limits_{j=1}^n x_j y_j =<x,y>
    And I haven't the slightest idea what (B) means, let alone how to prove it using A which I almost certainly did wrong.

    For (A): write both vectors as lin. comb. of the given orthonormal basis: x=\sum^n_{i=1}a_iv_i\,,\,\,y=\sum^n_{j=1}b_jv_j\,,  \,\,a_i,\,b_j\in\mathbb{C} , so:

    \left<x,y\right>=\left<\sum^n_{i=1}a_iv_i\,,\,\sum  ^n_{j=1}b_jv_j\right>=\sum^n_{i,j=1}a_i\overline{b  _j}\left<v_i,v_j\right> =\sum^n_{i=1}ai\overline{b_i} , as \left<v_i,v_j\right>=\delta_{i,j} .

    OTOH, \left<x,v_i\right>\overline{\left<y,v_i\right>}=\s  um^n_{k=1}a_k\left<v_k,v_i\right>\overline{\sum^n_  {k=1}b_k\left<v_k,v_i\right>} =a_i\overline{b_i} , and summing over i from 1 to n we get the same as above.

    Wat you did is wrong because one step before the last you write there the product of x_iy_{ij}v_{ij} and etc...what's this??


    About (B) also I don't understand what's going on: what's \phi_\beta , anyway? Though I suspect it is the canonical isomorphism from V\,\,\,to\,\,\,\mathbb{F}^n assigning to every vectors its coordinates wrt the basis \beta ...but then the exercise is pretty straightforward, after you fix the notation's mistakes.

    Tonio
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