Operators Whose Resolvents Have Convolution Representations and their Spectral Analysis

In this paper, we study spectral decompositions with respect to a system of generalized eigenvectors second-order differential operators on an interval whose resolvents possess convolution representations. We obtain the representation integral boundary conditions. Then, using generated by initial operator, construct Fourier transform. A connection between operation in original functional space and multiplication transforms is established. Finally, problem convergence expansions operator studied. Examples convolutions are also presented.