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Math Help - fourier transforms

  1. #1
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    fourier transforms

    Fact: every even Schwartz function on is the fourier transform of some integrable function.

    Using (or not!) that fact, find out if the following are the Fourier transforms of some integrable functions on R:

    A. , with

    B. , with

    C.

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  2. #2
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    Okay what is the definition of "even Schwartz function"? Are the given functions even Schwarz functions?
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  3. #3
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    A Schwartz function is a "rapidly decaying" function, in the sense that the function and all its derivatives decay faster than \frac{1}{x^n} for n = 1,2,3,4,...
    Schwartz space - Wikipedia, the free encyclopedia

    Those functions are certainly even, but none of them is Schwartz... the first two only decay as fast as \frac{1}{|y|^a}, and the last decays much slower than that (as \frac{1}{log|y|}). So, the Schwartz thing isn't useful here.
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