I am interested in clarifying the meaning and uses of the characteristic functions. I want to create a likelihood distribution in the Fourier Domain bearing in mind that it is a conditioned Poisson (or Gaussian) distribution in the inverse-Fourier domain.
1º) The characteristic function is defined as E[exp(itx)] and can be seen as the conjugate of the Fourier Transform of the density function f(x) if I am not wrong. If such a density function is the P(X=x) or P(a<X<b). Can I consider the characteristic function as the equivalent probabilities P(W=w) or P(A<W<B)?
I mean, is the characteristic function a kind of density function in the Fourier Domain for the distribution X?
2º) Suppose I define a likelihood function as a Poisson distribution.
Where the probability P(X=p|h) = exp(-h) [h^p]/p!
And I have the p dataset both in the Fourier (P) and non-Fourier domain (p).
What would be the equivalent expression for the charateristic function? or how can I get it?
At the beginning I thought I could say that, if the characteristic function for a Poisson distribution is exp[lambda(e^iw-1)], the equivalent expression could be something similar to exp[H(P-1)], bearing in mind that P are complex numbers, H as well. Or exp[|H|(e^i|P|-1) in order to keep lambda and w reals.
But now I believe I am misunderstanding things, especially related with point number 1. What is the role of the characteristic function?
If anybody can help would be great.