Inverting convolutions over finite intervals

Question

There are well-known techniques for inverting convolutions over the whole or half real line with Fourier and Laplace transformations, but on the face of it they can't be applied to an integral equation of the form:

$$\int_{-c}^c f(x-t)\rho(t) dt = g(x)\\ -c \le x \le c$$

where the unknown function to be determined is $\rho$, and $f$ will generally be non-zero over the full domain $-2c\le x-t\le2c$. In the examples I'm interested in, $f$ is an even function.

Is there a general strategy for solving for $\rho$?

Answer 1

A modification of the Wiener-Hopf method for this type of problems is described in Convolution equations on finite intervals and factorization of matrix functions and in Finite interval convolution operators with transmission property.