40 research outputs found

### Point evaluation and Hardy space on a homogeneous tree

We consider transfer functions of time--invariant systems as defined by
Basseville, Benveniste, Nikoukhah and Willsky when the discrete time is
replaced by the nodes of an homogeneous tree. The complex numbers are now
replaced by a C*-algebra built from the structure of the tree. We define a
point evaluation with values in this C*-algebra and a corresponding ``Hardy
space'' in which a Cauchy's formula holds. This point evaluation is used to
define in this context the counterpart of classical notions such as Blaschke
factors. There are deep analogies with the non stationary setting as developed
by the first author, Dewilde and Dym.Comment: Added references, changed notation

### Discrete Analytic Schur Functions

We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation problem

### Schur Analysis and Discrete Analytic Functions: Rational Functions and Co-isometric Realizations

We define and study rational discrete analytic functions and prove the existence of a coisometric realization for discrete analytic Schur multipliers

### On discrete analytic functions: Products, Rational Functions, and some Associated Reproducing Kernel Hilbert Spaces

We introduce a family of discrete analytic functions, called expandable
discrete analytic functions, which includes discrete analytic polynomials, and
define two products in this family. The first one is defined in a way similar
to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us
to define rational discrete analytic functions. To define the second product we
need a new space of entire functions which is contractively included in the
Fock space. We study in this space some counterparts of Schur analysis

### A new realization of rational functions, with applications to linear combination interpolation

We introduce the following linear combination interpolation problem (LCI):
Given $N$ distinct numbers $w_1,\ldots w_N$ and $N+1$ complex numbers
$a_1,\ldots, a_N$ and $c$, find all functions $f(z)$ analytic in a simply
connected set (depending on $f$) containing the points $w_1,\ldots,w_N$ such
that $\sum_{u=1}^Na_uf(w_u)=c.$ To this end we prove a representation
theorem for such functions $f$ in terms of an associated polynomial $p(z)$. We
first introduce the following two operations, $(i)$ substitution of $p$, and
$(ii)$ multiplication by monomials $z^j, 0\le j < N$. Then let $M$ be the
module generated by these two operations, acting on functions analytic near
$0$. We prove that every function $f$, analytic in a neighborhood of the roots
of $p$, is in $M$. In fact, this representation of $f$ is unique. To solve the
above interpolation problem, we employ an adapted systems theoretic
realization, as well as an associated representation of the Cuntz relations
(from multi-variable operator theory.) We study these operations in reproducing
kernel Hilbert space): We give necessary and sufficient condition for existence
of realizations of these representation of the Cuntz relations by operators in
certain reproducing kernel Hilbert spaces, and offer infinite product
factorizations of the corresponding kernels

### Discrete analytic Schur functions

We introduce the Schur class of functions, discrete analytic on the integer
lattice in the complex plane. As a special case, we derive the explicit form of
discrete analytic Blaschke factors and solve the related basic interpolation
problem

### Relative reproducing kernel Hilbert spaces

We introduce a reproducing kernel structure for Hilbert spaces of functions where differences of point evaluations are bounded. The associated reproducing kernels are characterized in terms of conditionally negative functions