On relations between analytic modular forms and Maass waveforms forPSL(2,Z)

In the standard Hilbert space L₂(F; dμ) of even automorphic functions, where F is the fundamental domain of the modular group PSL(2, ℤ), a full system of functions (in terms of classical analytic modular forms) is constructed, up to a finite-dimensional subspace. Therefore, in particular, almost every Maass waveform can be represented by a series in analytic forms. Bibliography: 9 titles. To O. A. Ladyzhenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 51–61. Translated by A. B. Venkov.