# Question regarding multidimensional integral

I would like to compute the following iterated integral in 3 dimesions: $$\int_{-\infty}^\infty \int_{-\infty}^\infty \left|\int_{-\infty}^\infty f(y)f(y-t)e^{-i 2 \pi t \xi} dy\right|dt d\xi$$ where $f$ is the spheroidal wave function which I can sample at arbitrary points using SpheroidalPS[].

The absolute value in the inner integral is the cause of the problem. What is the way of using NIntegrate to handle this absolute value for multiple integrals.

EDIT The requested code gives warnings

NIntegrate[Abs[NIntegrate[
SpheroidalPS[0, 0, Pi, y] SpheroidalPS[0, 0, Pi, y - t] Exp[-I 2 Pi t e],
{y, -Infinity, Infinity}]], {t, -Infinity, Infinity},
{e, -Infinity, Infinity}]

(*NIntegrate::inumr: The integrand ((-I) \[Infinity]) SpheroidalPS[0,0,\
[Pi],y] SpheroidalPS[0,0,\[Pi],-\[Infinity]] has evaluated to non-numerical
values for all sampling points in the region with boundaries {{-\
[Infinity],0.}}.*)

• Show us the code you used – MarcoB Jun 19 '18 at 15:47
• Hi Iconoclast, after asking 8 questions in Mathematica.SE you should know that here its considered helpful and polite show your own efforts and share your data and code attempts in a well formatted form, so we can quickly see the problem you are facing. Please help us to help you and edit your question accordingly. Also, it's never too late to take the tour. – rhermans Jun 19 '18 at 16:34
• It looks a lot like the Fourier transform of a convolution. – bill s Jun 19 '18 at 16:40
• @rhermans I have added the code. – Iconoclast Jun 19 '18 at 17:10
• @bill s its the L1 norm of the STFT (Short Time Fourier Transform) of a prolate with respect to the same prolate as a test function. – Iconoclast Jun 19 '18 at 17:10