Inverse of a matrix defined by a function

Question

I have a matrix $M$ whose elements are defined by some function $$M_{ij} = f ( |i-j| ) $$

Is it possible to derive a function which defines the elements of the matrix inverse $M^{-1}$ i.e.

$$M^{-1}_{ij} = g(i,j) $$

in terms of the original function $f ( |i-j| ) $ , either in general or for some particularly helpful form of $f$? I have some freedom over $f$ but it would be helpful for it to be a spline.

Thanks.

Answer 1

This is the problem of inverting a Toeplitz matrix. Its inverse is not Toeplitz in general. Search for "inverse of Toeplitz matrix" on the net.