On a New Class of Reproducing Kernel Spaces and a New Generalization of Iohvidov's Laws
A new class of finite dimensional reproducing kernel spaces of m × 1 vector valued analytic functions on a fairly general domain Ω+ is introduced. The reproducing kernels of these spaces have a special form which is based on an m × m matrix valued function Θ, which is J unitary on the boundary of Ω+. Every invertible Hermitian matrix can be interpreted as the Gram matrix of a suitably chosen basis in such a space, where m is equal to the appropriately defined displacement rank of the given matrix. Orthogonal decompositions of the space are developed in terms of Schur complements of the matrix and factorizations of the Θ, much as in the classical cases wherein Ω+ is either the unit disc or the half plane. Finally, a new generalization of the Iohvidov laws (which extends earlier generalizations by the same authors) is deduced as an application of the theory.
D. Alpay and H. Dym. On a new class of reproducing kernel spaces and a new generalization of Iohvidov's laws. In Linear Algebra and its Applications, vol. 178 (1993) 109-183.