Elementary solutions of a homogeneous $q$-sided convolution equation

Spectral synthesis on the complex plane related to solutions of some homogeneous equations of convolution type. There is a method to obtain solutions: we describe the elementary solutions set of the equation (spectral analysis) and prove the approximation theorem (spectral synthesis). In this paper we use the method for some homogeneous equations of convolution type, which appears from spectral synthesis problem for some differential operator.