I'm not sure if this will help, but I know some about taylor series and can do some fractional calculus (Where can't we do fractional calculus)

If is holo on and satisfies some fast growth at imaginary infinity and negative infinity. And if in the half plane. Then fix :

Maybe that might help some of you? The unfortunate part is as we're going to get decay to zero. I'm not sure about the iterates. We can also note this is a modified fourier transform and so we can apply some of Paley Wiener's theorems on bounding fourier transforms from the original functions. I.e: We can bound the taylor series by the function in the integral. Therefore maybe if we get very fast decay to zero we can talk about asymptotics of .

If is holo on and satisfies some fast growth at imaginary infinity and negative infinity. And if in the half plane. Then fix :

Maybe that might help some of you? The unfortunate part is as we're going to get decay to zero. I'm not sure about the iterates. We can also note this is a modified fourier transform and so we can apply some of Paley Wiener's theorems on bounding fourier transforms from the original functions. I.e: We can bound the taylor series by the function in the integral. Therefore maybe if we get very fast decay to zero we can talk about asymptotics of .