I would like to know what are the Fourier transform for the Gamma function and its logarithm.
if f(x)=gamma(x) and g(x)=log[gamma(x)]. What are the expression for F(w) and G(w)?
Furthermore, I would be interested in knowing how to translate density functions to Fourier domain. I guess the typical formula can be applied but I wonder if there is some basic bibliography or previous work that can be useful.
I am not mathematicien but enginner in astronomical problems. I am interested in creating a likelihood function in frequencial plane. In synthesis imaging in astronomy, sources are modeled like a poisson distribution for low emissions, so a likelihood function can be design to compare data obtained with a telescope and a model that is modified iteratively until convergence.
log L= SUMj[(-h + p*log(h) - log(p!)]
Where p are my datas and h my model in each iteration. Both p and h are 2D-vectors. Hence, looking for the maximums of that function you reach to a Richardson-Lucy scheme.
I can go from discrete poisson distribution to the continuous one by substituting the factorial at the denominator by a gamma function. I think this must be done because factorial is only defined for integers and, in Fourier domain, I would work with complex.
Finally, in order to work in the frequencial plane I think I should/could calculate the Fourier transform of the likelihood function.
This is a hard work and any help will be fruitful. I also think I could define some distribution directly in frequencial domain to describe my physical process (maybe an uniform, erlang, do not know, etc) but knowing how a poisson distribution looks like in Fourier domain will be helpful.
Thanks in advance.