  Properties of analytic vectors in representations of SL(2,R) are used to give
new bounds for the triple products recently considered by P. Sarnak. A
conjecture of Sarnak about such products is proved. The results of this paper
generalize results of A. Good and M. Jutila about special cases, but the
techniques are entirely different. One consequence of these results is a new
estimate of the magnitude of the Fourier coefficients of cusp forms for
non-arithmetic sub-groups of SL(2,R).
