On Riccati Equations and Reproducing Kernel Spaces
Harry Dym · Birkhäuser Basel eBooks · 2001
A class of finite dimensional reproducing kernel Krein spaces of vector valued rational functions M X with an indefinite inner product that is defined in terms of a singular Hermitian matrix X is analyzed. It is shown that if X is positive semidefinite, then M X is a reproducing kernel Hilbert space of the kind that originates in the work of L. de Branges if and only if X is a solution of an associated Riccati equation and a certain invariance condition (which is automatically fulfilled in some cases) is met. Analogous conclusions are obtained for the case when X is Hermitian and M X is a Krein space. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.