Abstract

Let f(q(tau), q(z)) = Sigma(n, r)c(n, r) q(tau)(n)q(z)(r) be a power series whose coefficients satisfy a par ticular periodicity condition depending on the integer r module 2m. We first associate to f(q(tau), q(z)) a 2m-vector-valued function Lambda(f, s) via a generalized Mellin transform. Then we show that the function Lambda(fs) is entire, bounded on vertical strips and satisfies certain matrix functional equation if, and only if, f(q(tau), q(z)) is the Fourier expansion of a Jacobi cusp form of index m invariant under the group SL(2, Z) x Z(2). This is the direct analogue of Hecke's converse theorem for elliptic cusp forms in the context of Jacobi cusp forms on SL(2, Z) x Z(2). (C) 1996 Academic Press. Inc.