How to expand the Gaussian Kernel into series of eigenfunctions?

I have a gaussian kernal

How to expand it into a series of eigenfunctions ?Any help will be appreciated.

Answer 1

[Vladimir Sovkov]

Fitst of all, you should specify --nothing can be done with an arbtrary function. Then, you can try to calculate the integrals analytically using the call function "int" in the Symbolic Math Toolbox, but there is no guarantee that an analytical solution exists at all. To find the numerical solution you should compute the integrals numerically at every x, y, i of interest; use the call function "inegral" for this purpose. There are special cases where the computetion can be simplifyed, e.g., in a case you mean the Fourier transfer (you are expressed in terms of complex exponents or sin/cos functions), the call function "fourier" (symbolic) or "fft" and "ifft" (numerical) can be used. The new non-uniform Fourier transform functions "nufft" and "nufftn" are announced to be introduced in Matlab 2020a.

[saadi]

Thank you for kind response. Its very helpful but Is it possible for you to send me some links or tutorials related to this question. Any programming language is fine. Thanks in advance for your time and support!

[Vladimir Sovkov]

If you want to learn about the computational functions, just search Matlab Help system for the functions I listed above.

If you want to learn about the mathematics behind your problem (I do not know what your educational background is ...), read any textbook on Functional Analysis; a brief introduction in Wikipedia is, e.g., at https://en.wikipedia.org/wiki/Functional_analysis

[Vladimir Sovkov]

I misprinted in the 1st part of my answer. Must be "integral" but not "inegral".