Nevanlinna-Pick Kernels and Localization

We describe those reproducing kernel Hilbert spaces of holomorphic functions on domains in ${\Bbb C}^d$ for which an analogue of the Nevanlinna-Pick theorem holds, in other words when the existence of a (possibly matrix-valued) function in the unit ball of the multiplier algebra with specified values on a finite set of points is equivalent to the positvity of a related matrix. Our description is in terms of a certain localization property of the kernel.
Comment: in Proceedings of 17th International Conference on Operator Theory at Timisoara, 1998, Theta Foundation, Bucharest