Reproducing kernel Hilbert spaces, linear (deterministic and stochastic) systems and inverse scattering

Functions analytic and contractive in the open unit disk play a key role in a host of mathematical domains, for instance operator models, inverse scattering and linear systems. We first discuss the main points in this circle of ideas. The reproducing kernel Hilbert spaces of the kind introduced by de Branges and Rovnyak are an important tool in the arguments.

In the second part of the talk we study the stochastic analogs of the situation above. The complex numbers are then replaced by an algebra of stochastic distributions, with a very special structure. it is the dual of a nuclear Fréchet space, and a countable union of increasing Hilbert spaces with decreasing norms, the norms being related by a family of inequalities.

Finally we briefly outline the free analog of these spaces of stochastic distributions.

References:

D. Alpay. The Schur algorithm, reproducing kernel spaces and system theory, volume 5, SMF/AMS Texts and Monographs of the Société mathématique de France (2001).

D. Alpay and I. Gohberg. Discrete systems and their characteristic spectral functions. Mediterrean Journal of Mathematics, vol. 4 (2007), pp. 1-32.

D. Alpay, P. Jorgensen and D. Levanony. A class of Gaussian processes with fractional spectral measures. Journal of Functional Analysis, Volume 261 (2011), pp. 507-541.

D. Alpay and G. Salomon. Non-commutative stochastic distributions and applications to linear systems theory. Stochastic Processes and their Applications, vol. 123 (2013), pp. 2303-2322.

D. Alpay, P. Jorgensen and G. Salomon: On free stochastic processes and their derivatives. Stochastic Processes and their Applications, vol. 214 (2014), 3392-3411.