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Math Help - Set of continuous functions dense in C[-pi,pi]

  1. #1
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    Set of continuous functions dense in C[-pi,pi]

    How do i show that the set of continuous functions in C[-\pi,\pi] which are periodic, that is f(+\pi)=f(-\pi) is dense in L^{2}[-\pi,pi]
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  2. #2
    Grand Panjandrum
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    Re: Set of continuous functions dense in C[-pi,pi]

    Quote Originally Posted by Chandru1 View Post
    How do i show that the set of continuous functions in C[-\pi,\pi] which are periodic, that is f(+\pi)=f(-\pi) is dense in L^{2}[-\pi,\pi]
    Consider the convergence of Fourier series of functions in L^{2}[-\pi,\pi]?

    (since the partial sums of the Fourier series are in C[-\pi,\pi] )

    CB
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    Re: Set of continuous functions dense in C[-pi,pi]

    Quote Originally Posted by CaptainBlack View Post
    Consider the convergence of Fourier series of functions in L^{2}[-\pi,\pi]?

    (since the partial sums of the Fourier series are in C[-\pi,\pi] )

    CB

    Can you give a full proof of this. I am not aware of anything. A reference page in an internet would be good.
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  4. #4
    Grand Panjandrum
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    Re: Set of continuous functions dense in C[-pi,pi]

    Quote Originally Posted by Chandru1 View Post
    Can you give a full proof of this. I am not aware of anything. A reference page in an internet would be good.

    Googleing for "l2 convergence of fourier series" will give you several PDFs which discuss this, alternativly follow the links in the Norm Convergence area of the Wikipedia page:

    Convergence of Fourier series - Wikipedia, the free encyclopedia

    CB
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