Holomorphic functions with positive real part on polycylinders

The purpose of this paper is to generalize the classical Riesz-Herglotz integral representation and Pick-Nevanlinna interpolation theorems to functions of several complex variables. A generalization of these results in a somewhat different direction has already been given in [2]; the functions with positive real part, which will be the objects of our study here, turn out to be closely related to a subclass of the class Hm considered in [2]. We note that some of our results (Theorem 1 and the necessity part of Theorem 2) apply to a much larger class of domains than the polycylinders. In fact, the Szegö kernel function, which plays a crucial role in our investigation, exists and has very similar properties in all the bounded homogeneous domains which are starlike, circular, and whose isotropy group is linear and transitive on the BergmanSilov boundary (cf. [1] ; the symmetric domains of E. Cartan are all such). Our proofs apply to this more general situation as well, with the only change that the system (1) has to be replaced by a system of polynomials orthonormal on the Bergman-Silov boundary and constructed from the irreducible representations of the isotropy group. The holomorphic elements of this system will then span the range of the projection P. Our method of proving the second part of Theorem 2, however, seems not to be applicable here, and so the problem of the sufficiency of our conditions in the general case remains open. We shall denote by z = (z1,---,zm) the points of complex Euclidean m-space Cm. D will be the unit polycylinder of C'\