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Math Help - Proofs about convolutions

  1. #1
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    Proofs about convolutions

    The problem has 5 parts and is based on:


    Let f,g \in L^2(T, dx). Define the convolution of the two functions by: (f ** g)(x) = 1/(2pi) \int_T f(x-t)g(t)dt


    a) Prove that f**g is a continuous function.
    b) Prove that f**g = g**f
    c) Denote by h(n) = 1/(2pi) \int_Th(x)e^{-inx}dx the Fourier coefficients of a function h \inL^2(T, dx). Prove that f(*^*g)(n) = \hat{f}(n)\hat{g}(n), n \inZ
    d) Deduce that, for three functions in L^2: f**(g**h) = (f**g)**h.
    e) Is there a function e \in L^2(T, dx) with the property: e*f = f, f \inL^2(T,dx)?


    Where ** is the convolution symbol and T = {z such that |z| = 1} = { e^{it} | t \in[0, 2pi]}


    Any help at all will be amazing. I have no idea what is going on.
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  2. #2
    Super Member girdav's Avatar
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    Re: Proofs about convolutions

    For the first question use density of the continuous functions. For the second one, a substitution, using translation invariance.
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  3. #3
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    Re: Proofs about convolutions

    For the second question, do u mean something like:
    t --> (x-t)
    (x-t) --> t x - (x-t) -t
    ?
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  4. #4
    Super Member girdav's Avatar
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    Re: Proofs about convolutions

    Yes, that's what I mean.
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