9,105 research outputs found
Schur functions and their realizations in the slice hyperholomorphic setting
we start the study of Schur analysis in the quaternionic setting using the
theory of slice hyperholomorphic functions. The novelty of our approach is that
slice hyperholomorphic functions allows to write realizations in terms of a
suitable resolvent, the so called S-resolvent operator and to extend several
results that hold in the complex case to the quaternionic case. We discuss
reproducing kernels, positive definite functions in this setting and we show
how they can be obtained in our setting using the extension operator and the
slice regular product. We define Schur multipliers, and find their co-isometric
realization in terms of the associated de Branges-Rovnyak space
Stability is realization-dependent: some examples
This paper gives some examples with the same impulse response, both approximately controllable and observable, but one of them is exponentially stable and the other is unstable. Some related spectral properties are also investigated
Rational inner functions in the Schur-Agler class of the polydisk
Every two variable rational inner function on the bidisk has a special
representation called a transfer function realization. It is well known and
related to important ideas in operator theory that this does not extend to
three or more variables on the polydisk. We study the class of rational inner
functions on the polydisk which do possess a transfer function realization (the
Schur-Agler class) and investigate minimality in their representations.
Schur-Agler class rational inner functions in three or more variables cannot be
represented in a way that is as minimal as two variables might suggest.Comment: 14 page
Minimal symmetric Darlington synthesis
We consider the symmetric Darlington synthesis of a p x p rational symmetric
Schur function S with the constraint that the extension is of size 2p x 2p.
Under the assumption that S is strictly contractive in at least one point of
the imaginary axis, we determine the minimal McMillan degree of the extension.
In particular, we show that it is generically given by the number of zeros of
odd multiplicity of I-SS*. A constructive characterization of all such
extensions is provided in terms of a symmetric realization of S and of the
outer spectral factor of I-SS*. The authors's motivation for the problem stems
from Surface Acoustic Wave filters where physical constraints on the
electro-acoustic scattering matrix naturally raise this mathematical issue
- …