Convolutions. Spectral decompositions over different vertices (Q1375224)

The author [J. Math. Sci., New York 83, 731-744 (1997); translation from Zap. Nauchn. Semin. POMI 212, 71-90 (1994; Zbl 0870.11055)] obtained the spectral decomposition of the convolution over the vertex \(\alpha = \infty\) of the fundamental domain of the congruence subgroup \(\Gamma _{0}(q)\). The present paper solves the problem in the case \(q>1\) for the vertex \(\alpha=0\). This result allows for a comparison of these two decompositions of one and the same convolution and for a study of the spectral structure and spectral coefficients.