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 $\mathscr{F}_{\mathbb{R^{+}}}$, $$(\mathscr{F}_{\mathbb{R^{+}}}x)(t)=\frac{1}{\sqrt{2\pi}} \int\limits_{\mathbb{R^{+}}}x(\xi)e^{it\xi}\,d\xi\,,\ \ \ t\in{}{\mathbb{R^{+}}},$$ is studied. The operator $\mathscr{F}_{\mathbb{R^{+}}}$ is considered as an operator acting in the space $L^2(\mathbb{R^{+}})$. The functional model for the operator $\mathscr{F}_{\mathbb{R^{+}}}$ is constructed. This functional model is the multiplication operator on the appropriate $2\times2$ matrix function acting in the space $L^2(\mathbb{R^{+}})\oplus{}L^2(\mathbb{R^{+}})$. Using this functional model, the spectrum of the operator $\mathscr{F}_{\mathbb{R^{+}}}$ is found. The resolvent of the operator $\mathscr{F}_{\mathbb{R^{+}}}$ 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 $2\times2$ matrix $G$ 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 $n$, the family $\big\{L_n(z,G)\big\}_G$, where $G$ runs over the set of all $2\times2$ matrices, is a three-parametric family. A natural parametrization of this family is found. The polynomial $L_n(z,G)$ is expressed in terms of these parameters and the Chebyshev polynomial $T_n$. The zero set of the polynomial $L_n(z,G)$ is described.Comment: 15 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 measures which generate the scalar product in a space of rational functions

Let $z_1,z_2,\,\ldots\,,z_n$ be pairwise different points of the unit disc and $\mathscr{L}(z_1,z_2,\,\ldots\,z_n)$ be the linear space generated by the rational fractions $\frac{1}{t-z_1} , \frac{1}{t-z_2} , \cdots\ , \frac{1}{t-z_n}\cdot$ Every non-negative measure $\sigma$ on the unit circle $\mathbb{T}$ 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 $\sigma$ 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 $m$ is the normalized Lebesgue measure on $\mathbb{T}$.Comment: 9 page

On summation of the Taylor series of the function 1/(1-z) by the theta summation method

The family of the Taylor series f_{\epsilon}(z)= \sum\limits_{0\leq{}n<\infty}e^{-\epsilon n^2}z^n is considered, where the parameter \epsilon, which enumerates the family, runs over ]0,\infty[. The limiting behavior of this family is studied as \epsilon\to+0.Comment: 19 pages, 3 figure

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

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 $\nu_0(t),\nu_1(t),\,\ldots\,,\nu_n(t)$ be the roots of the equation $R(z)=t$, where $R(z)$ is a rational function of the form $$R(z)=z+\sum\limits_{k=1}^n\frac{\alpha_k}{z-\mu_k},$$ $\mu_k$ are pairwise different real numbers, $\alpha_k>0,\,1\leq{}k\leq{}n$. Then for each real $\xi$, the function $e^{\xi\nu_0(t)}+e^{\xi\nu_1(t)}+\,\cdots\,+e^{\xi\nu_n(t)}$ is exponentially convex on the interval $-\infty<t<\infty$.Comment: 9 page

The function (cosh⁡at2+b)(\cosh\sqrt{at^2+b}) is exponentially convex

Given positive numbers $a$ and $b$, the function $\sqrt{at^2+b}$ is exponentially convex function of $t$ on the whole real axis. Three proofs of this result are presented.Comment: 20 page