A method of parametric solution of convolution equations (Q1708347)

The paper considers the approximate numerical solution of a convolution-type integral equation \[ \int_{-\infty}^\infty K(t-s) z(s) \,ds = u(t), \] where \(K\) and \(u\) are known, and \(z(t)\) is an unknown function. It is assumed that \(z(t)\) belongs to a family of functions described by a small number of parameters. Of special interest are the gamma model \[ z(t;\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} t^{\alpha-1} e^{-\beta t}, \] and the shifted gamma model \[ z(t;\alpha,\beta,\tau) = \frac{\beta^\alpha}{\Gamma(\alpha)} (t-\tau)^{\alpha-1} e^{-\beta (t-\tau)}. \] Relationships between the moments of \(K\), \(z\), and \(u\) lead to a system of equations for the parameters, which can then be solved numerically. The author points out that this approach can also be used in the Tikhonov regularization, by using the parametric solution to estimate the regularization parameter \(\delta\).