 Cauchy convolution semigroup

The normalized Cauchy distribution densities $K_a$, $a>0$, form a convolution semigroup: convolution adds their scale parameters. This follows from the convolution theorem, since their Fourier transforms are proportional to $e^{-a|k|}$. Consequently an integral equation with a Cauchy kernel can be solved by subtracting scale parameters when the resulting scale is positive.