44 research outputs found
Right and Left Joint System Representation of a Rational Matrix Function in General Position (System Representation Theory for Dummies)
For a rational matrix function R of one variable in general position, the
matrix functions R(x)/R(y) and R(y)\R(x) of two variables are considered. For
these matrix functions of two variables, representations which are analogous to
the system representations (or realizations) of a rational matrix function of
one variable are constructed. This representation is called the joint right
[the joint left] system representation
A functional model for the Fourier--Plancherel operator truncated on the positive half-axis
The truncated Fourier operator , is studied. The operator
is considered as an operator acting in the space
. The functional model for the operator
is constructed. This functional model is the
multiplication operator on the appropriate matrix function acting in
the space . Using this
functional model, the spectrum of the operator
is found. The resolvent of the operator is
estimated near its spectrum.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1208.381
On a family of Laurent polynomials generated by 2x2 matrices
To a matrix with complex entries, we relate the sequence of
Laurent polynomial L_n(z,G)=\tr \big(G\big[\begin{smallmatrix}z&0\\
0&z^{-1}\end{smallmatrix}\big]G^{\ast}\big)^n. It turns out that for each
, the family , where runs over the set of all
matrices, is a three-parametric family. A natural parametrization of
this family is found. The polynomial is expressed in terms of these
parameters and the Chebyshev polynomial . The zero set of the polynomial
is described.Comment: 15 page
Self-adjoint boundary conditions for the prolate spheroid differential operator
We consider the formal prolate spheroid differential operator on a finite
symmetric interval and describe all its self-adjoint boundary conditions. Only
one of these boundary conditions corresponds to a self-adjoint differential
operator which commute with the Fourier operator truncated on the considered
finite symmetric interval.Comment: 10 page
On measures which generate the scalar product in a space of rational functions
Let be pairwise different points of the unit disc
and be the linear space generated by the
rational fractions Every non-negative measure on the unit circle
generates the scalar product
\langle\,f\,,\,g\,\rangle_{\!_{L^2_\sigma}}
=\int\limits_{\mathbb{T}}f(t)\,\bar{g(t)}\,\sigma(dt), \quad
\forall\,f,g\,\in\,L^2_\sigma. The measures are described which
satisfy the condition \langle\,f\,,\,g\,\rangle_{\!_{L^2_\sigma}}=
\langle\,f\,,\,g\,\rangle_{\!_{L^2_m}},\quad
\forall\,f,g\in\mathscr{L}(z_1,z_2,\,\ldots\,z_n), where is the
normalized Lebesgue measure on .Comment: 9 page
Steiner-Minkowski Polynomials of Convex Sets in High Dimension, and Limit Entire Functions
For a convex set (K) of the (n)-dimensional Euclidean space, the
Steiner-Minkowski polynomial (M_K(t)) is defined as the (n)-dimensional
Euclidean volume of the neighborhood of the radius (t). Being defined for
positive (t), the Steiner-Minkowski polynomials are considered for all complex
(t). The renormalization procedure for Steiner polynomial is proposed. The
renormalized Steiner-Minkowski polynomials corresponding to all possible solid
convex sets in all dimensions form a normal family in the whole complex plane.
For each of the four families of convex sets: the Euclidean balls, the cubes,
the regular cross-polytopes and the regular symplexes of dimensions (n), the
limiting entire functions, as (n) tends to infinity, are calculated explicitly.Comment: 15 page
On the roots of a hyperbolic polynomial pencil
Let be the roots of the equation
, where is a rational function of the form
are pairwise
different real numbers, . Then for each real
, the function
is exponentially
convex on the interval .Comment: 9 page
The function is exponentially convex
Given positive numbers and , the function is
exponentially convex function of on the whole real axis. Three proofs of
this result are presented.Comment: 20 page
On Transformation of Potapov's Fundamental Matrix Inequality
According to V.P.Potapov, a classical interpolation problem can be
reformulated in terms of a so-called Fundamental Matrix Inequality (FMI). To
show that every solution of the FMI satisfies the interpolation problem, we
usualy have to transform the FMI in some special way. In this paper the number
of of transformations of the FMI which come into play are motivated and
demonstrated by simple, but typical examples
On the completeness of Gaussians in a Hilbert functional space
The completeness of Gaussians in a Hilbert functional space is establishedComment: 23 page