2
$\begingroup$

Let $\mathcal{F}$ denotes the Fourier transform.

It is known that $\mathcal{F}(e^{-4\pi^2 i t |x|^2})(\xi)= e^{i |\xi|^2/4t}{(4\pi i t)^{-n/2}} \ (x, \xi\in \mathbb R^n).$

My question is: what is $\mathcal{F}(e^{-4\pi^2 i t |x|^\alpha})(\xi)$ for some $\alpha >0, \alpha \neq 2$?

Motivation: This kernel appears in the evolution of fractional Schrödinger equation.

$\endgroup$
1
$\begingroup$

This is known as the Kohlrausch-Williams-Watts function. There is no exact closed-form expression.

The numerical evaluation for $n=1$ has been studied in Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation libkww (2009).

There exists an analytic approximation, described in Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions (1991).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.