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Math Help - L1-norm approximation

  1. #1
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    L1-norm approximation

    Dear friends,

    Do you think it is possible to approximate the L1-norm of a finite discrete sequence {x[n]} (i.e., ||x|| = sum(|x[n]|) ) given the values of its discrete Fourier transform? If not, is there a theorem asserting that, in this case, it is impossible to arrive at a "Parseval-like" (approximative) relation?

    Thank you very much in advance for your responses.
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  2. #2
    Super Member Rebesques's Avatar
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    If I remember right, the fourier transform of an L^1 sequence, need not belong in L^1. So the norm would be meaningless
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Rebesques
    If I remember right, the fourier transform of an L^1 sequence, need not belong in L^1. So the norm would be meaningless
    The sequence is finite, so we are dealing with the Discrete FT here.
    As the DFT of a bounded finite sequence is bounded it is of necessity
    L^1.

    RonL
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  4. #4
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    Yes, this is the case. For simplicity (and in order to avoid the "definiteness issue") I assumed the sequence to be finite. In this case, its Fourier transform is bounded and hence well-defined. Moreover, if the norm in question were the L2 norm, the Parseval theorem guarantees that the L2 norm of the sequence is equal to that of its (discrete) Fourier coefficients. However, it is possible to figure out (even approximately) what is the L1 norm of the sequence given only the Fourier coefficients of the latter?

    Thank you very much for your interest and responses.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Neighbor
    Yes, this is the case. For simplicity (and in order to avoid the "definiteness issue") I assumed the sequence to be finite. In this case, its Fourier transform is bounded and hence well-defined. Moreover, if the norm in question were the L2 norm, the Parseval theorem guarantees that the L2 norm of the sequence is equal to that of its (discrete) Fourier coefficients. However, it is possible to figure out (even approximately) what is the L1 norm of the sequence given only the Fourier coefficients of the latter?
    In principle it is. This is because the DFT of a finite sequence contains all the information
    embodied in the original sequence, and infact the inverse DFT will reconstruct the
    orginal exactly.

    However I'm not aware of any simple elegant equivalent of Parseval's
    theorem for the L1 norm, but that does not mean that there isn't one

    RonL
    Last edited by CaptainBlack; January 20th 2006 at 12:50 PM.
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