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Math Help - Some Fourier Analysis

  1. #1
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    Some Fourier Analysis

    I have to proof the following:

    If f:\mathbb{T}\rightarrow\mathbb{C} Riemann integrable and g:\mathbb{T}\rightarrow\mathbb{C} continuous, then:

    \lim_{n\to\infty}\frac{1}{2\pi}\int_{0}^{2\pi}f(\t  heta)g(n\theta)d\theta = \hat{f}(0)\hat{g}(0)

    Where a Fourier coefficient \hat{f}(n) is given by \frac{1}{b-a}\int_{a}^{b}f(x)e^{-2\pi inx/L}dx, \qquad n \in \mathbb{Z}

    A hint is given to first proof it for g a trigonometric function and then in general for continuous functions.


    What might help in solving this is a property of convolutions. Given two 2\pi-periodic Riemann integrable functions f and g on \mathbb{R}, then their convolution f\ast g on [-\pi,\pi] is defined by
    (f\ast g)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(y)g(x-y)dy
    Our property of (probable) interest is: suppose f and g are 2\pi-periodic Riemann integrable functions. Then:
    \widehat{f\ast g}=\hat{f}(n)\hat{g}(n).


    I'm rather clueless on how to prove this one..
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  2. #2
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    Quote Originally Posted by brouwer View Post
    I have to proof the following:

    If f:\mathbb{T}\rightarrow\mathbb{C} Riemann integrable and g:\mathbb{T}\rightarrow\mathbb{C} continuous, then:

    \lim_{n\to\infty}\frac{1}{2\pi}\int_{0}^{2\pi}f(\t  heta)g(n\theta)d\theta = \hat{f}(0)\hat{g}(0)

    Where a Fourier coefficient \hat{f}(n) is given by \frac{1}{b-a}\int_{a}^{b}f(x)e^{-2\pi inx/L}dx, \qquad n \in \mathbb{Z}

    A hint is given to first proof it for g a trigonometric function and then in general for continuous functions.


    What might help in solving this is a property of convolutions. Given two 2\pi-periodic Riemann integrable functions f and g on \mathbb{R}, then their convolution f\ast g on [-\pi,\pi] is defined by
    (f\ast g)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(y)g(x-y)dy
    Our property of (probable) interest is: suppose f and g are 2\pi-periodic Riemann integrable functions. Then:
    \widehat{f\ast g}=\hat{f}(n)\hat{g}(n).


    I'm rather clueless on how to prove this one..
    You should try to use the hint. If g is a constant, this is trivial. Assume now that g(x)=e^{ikx} for some k\in\mathbb{N}^*. Then \frac{1}{2\pi}\int_0^{2\pi} f(\theta)g(n\theta)d\theta = \widehat{f}(-kn), and maybe you know that the Fourier coefficients in this case converge to 0 at infinity (this is also called Lebesgue's lemma) hence you find that the limit is zero, which is ok since \widehat{g}(0)=0.

    Then use uniform approximation of a general function g by trigonometric ones, i.e. linear combinations of functions like the previous one (it comes from Stone-Weierstrass theorem).
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  3. #3
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    Thanks for your reply, a great help and insight!

    One quick question though, what do you mean by \mathbb{N}^*? Never mind, I figured it's just \mathbb{N}_{>0}


    Thanks I got it completely now!

    Cheers.
    Last edited by brouwer; April 16th 2010 at 05:53 AM.
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