#### Milestone Year

### 1988

#### Reproducing Kernel Spaces

Since the early days of the Department of Mathematics in the 1970's, a significant effort by one or more department members, numerous students, postdocs and senior visitors has been invested in the study of assorted reproducing kernel spaces and their applications.

In the beginning attention focused largely on reproducing kernel Hilbert spaces of holomorphic and meromorphic functions on the open unit disc, or the open upper half plane or the full complex plane.

Early applications were to trace formulas, prediction theory of stationary stochastic processes and inverse problems for some second order differential equations.

Later research extended these investigations to vector valued reproducing kernel Hilbert spaces and Krein spaces and extended the applications to include bitangential interpolation problems, direct and inverse problems for canonical systems of integral and differential equations and Dirac-Krein systems, prediction for vector valued weakly stationary processes and zero location problems.
Research in many of these areas is still being actively pursued today.

Transparent models were developed for finite dimensional reproducing kernel spaces. One particularly pleasing that was discovered in the late eighties identified a mysterious identity due to L. de Branges that served to partially characterize an important class of reproducing kernel Hilbert spaces as the spaces in which the inner product is based on an invertible Hermitian matrix P that is a solution of a related Lyapunov-Stein equation. Subsequent generalizations characterized sesqui-linear forms based on singular matrices P with solutions of Riccati equations. Enroute, some useful connections with state space realization formulas were obtained