1. ## 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.

2. Originally Posted by mbbx5va2
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

3. ## Thanks

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)))