Analytical convolution scheme using eigenfunctions of the Fourier Transform

[krzystyniak]

An idea for a fast analytical convolution of two signals:

1. The Hermite polynomial series multiplied by a Gaussian kernel (Gram-Charlier expansions) are eigenfunctions of the FT;
2. One can expand two signals to be convolved in the GC series;
3. One can obtain a closed analytical formula for the multiplication of the FT of the GC series.
4. Finally, another formula can be obtained for the IFT of the multiplied FTs of the GC series.

Convolution_using_GC.zip

I have prepared some scripts in MATLAB and Python. They should be correct syntax-wise (I used a bit of ChatGPT for that), but they are proving to be very slow in the demo mode...
Food for thought...

[tkittel]

Thanks, always interesting with new ideas for faster convolutions!