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Math Help - Question about lebesgue space and fourier transforms

  1. #1
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    Question about lebesgue space and fourier transforms

    Hi, i need some help with this problem. The material is from a course called 'Numerical functional analysis'. if f(x)=u(x)*exp{-alpha*x} for alpha >0. Where u(x) is the step function ie u(x)=0 for x<0 and u(x)=1 for x larger than or equal to 0. Then does f prime (ie 1st derivative of f) belong to lebesgue space with p=2 over the real line.
    Earlier parts of the question ask to calculate fourier transform for f and f prime. I am thinking along the lines of plancherel's theorem. This would mean i would have to show f prime is a fourier transform which can be checked using the inverse fourier transform formula to find the original function. However then i would have to prove this function belongs to the intersection of lebesgue space p=1 and lebesgue space p=2 both over real line. (this is part of plancherel's theorem) Any suggestions would be much appreciated thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by mbbx5va2 View Post
    Hi, i need some help with this problem. The material is from a course called 'Numerical functional analysis'. if f(x)=u(x)*exp{-alpha*x} for alpha >0. Where u(x) is the step function ie u(x)=0 for x<0 and u(x)=1 for x larger than or equal to 0.
    f' is the sum of a delta functional and a piece-wise continuous function, which is a distribution and not in any L^p space, however it is integrable in the sense of distributions (though probably not square integrable) which is all that one needs for the I/FT defined on appropriate spaces of distributions.


    Then does f prime (ie 1st derivative of f) belong to lebesgue space with p=2 over the real line.
    Earlier parts of the question ask to calculate fourier transform for f and f prime. I am thinking along the lines of plancherel's theorem. This would mean i would have to show f prime is a fourier transform which can be checked using the inverse fourier transform formula to find the original function. However then i would have to prove this function belongs to the intersection of lebesgue space p=1 and lebesgue space p=2 both over real line. (this is part of plancherel's theorem) Any suggestions would be much appreciated thanks.
    CB
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  3. #3
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    Thanks

    Thanks for your reply CaptainBlack it was most useful.
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  4. #4
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    so why cant a distribution be in L2???
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by johnbarkwith View Post
    so why cant a distribution be in L2???
    Well it can if it is also a function (slight abuse of terminology here), but in this case we have distributions that are not functions in the conventional sense, \delta and \delta ' are definitely not in L^2.

    (you will note that I described the entity in question as "a distribution and not in any L^p space", this is not a claim that an arbitary distribution cannot be in an L^p space (with an suitable fuzzyness about what it means for a distribution to be a function, in fact all functions in L^p are distributions under an appropriate interpretation (that is their integrals with functions from the space of test functions exist)))

    CB
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