During applied mathematics, I am wondering how to decompose the data $[x,y]$ into several elliptic-shaped functions, namely
$$f[x,y]= H \cdot \left( 1 + \bigl( \frac{x-x_1}{a_1} \bigr)^2 + \bigl( \frac{y-y_1}{b_1} \bigr)^2 \right)^{-3/2}$$
Is there any theorems or methods about it?
or is it possible to construct an inner product space whose basis functions are Lorentzian functions (or Gaussian)? (Maybe orthonormality doesn't satisfy with classical inner product.)
Any ideas are welcome! Thank you very much.