An Interpolation Problem for Functions with Values in a Commutative Ring

Abstract

It was recently shown that the theory of linear stochastic systems can be viewed as a particular case of the theory of linear systems on a certain commutative ring of power series in a countable number of variables. In the present work we study an interpolation problem in this setting. A key tool is the principle of permanence of algebraic identities.1991 Mathematics Subject Classification. 60H40, 93C05. Key words and phrases. white noise space, stochastic distributions, linear systems on rings.D. Alpay thanks the Earl Katz family for endowing the chair which supported his research.

On the extension of positive definite kernels to topological algebras

“…Moreover, extensions given by Eq. (1.2) contrasts with the one done, for instance, in the study of white noise space, where, with exception of 5 , the complex coefficients of the power series (and not the variable) in Eq. (1.1) are replaced by elements that take value in the space of stochastic distributions S −1 .…”

On the extension of positive definite kernels to topological algebras

“…Moreover, extensions given by Eq. (1.2) contrasts with the one done, for instance, in the study of white noise space, where, with exception of [1], the complex coefficients of the power series (and not the variable) in Eq. (1.1) are replaced by elements that take value in the space of stochastic distributions S −1 .…”

“…Theorem 3.2. Let K(z, w) be a K p×p -valued function, entire in z and w. Then: (1) For every pair (z, w) ∈ K 2 , the semi-infinite block matrix…”

On free stochastic processes and their derivatives

“…Together with the white noise space these two spaces form the Gelfand triple (S 1 , W, S −1 ), which plays a key role in the stochastic analysis in [20], and in the theory of stochastic linear systems and stochastic integration developped in [6,2,3,5,1]. The reason of the importance of this triple is the following result, see [20], which allows to work locally in a Hilbert space setting.…”