Fourier transform method for convolution equations

Summary

The Fourier transform method solves a convolution integral equation on the whole line or on $\R^n$ by passing to the frequency side. The convolution theorem turns the convolution with the kernel into multiplication by the transform of the kernel. So the integral equation becomes an algebraic equation for the transform of the unknown. One solves it by division and returns with the inverse Fourier transform. The method also shows when the equation has exactly one solution in $L^2$ for every right side in $L^2$: the symbol $1 - \lambda \hat k(\xi)$ must not vanish.