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Victor Katsnelson

Publications and source records attributed to Victor Katsnelson.

At least 19 recordsLinked to original sources
arXiv2018

A functional model for the Fourier--Plancherel operator truncated on the positive half-axis

The truncated Fourier operator $\mathscr{F}_{\mathbb{R^{+}}}$, $$ (\mathscr{F}_{\mathbb{R^{+}}}x)(t)=\frac{1}{\sqrt{2π}} \int\limits_{\mathbb{R^{+}}}x(ξ)e^{itξ}\,dξ\,,\ \ \ t\in{}{\mathbb{R^{+}}}, $$ 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.

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arXiv2016

The matrix function $e^{tA+B}$ is representable as the Laplace transform of a matrix measure

Given a pair $A,B$ of matrices of size $n\times n$, we consider the matrix function $e^{At+B}$ of the variable $t\in\mathbb{C}$. If the matrix $A$ is Hermitian, the matrix function $e^{At+B}$ is representable as the bilateral Laplace transform of a matrix-valued measure $M(dλ)$ compactly supported on the real axis: $$e^{At+B}=\int{}e^{λt}\,M(dλ).$$ The values of the measure $M(dλ)$ are matrices of size $n\times n$, the support of this measure is contained in the convex hull of the spectrum of $A$. If the matrix $B$ is also Hermitian, then the values of the measure $M(dλ)$ are Hermitian matrices. The measure M(dλ) is not necessarily non-negative.

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arXiv2016

On a special case of the Herbert Stahl theorem

The BMV conjecture states that for $n\times n$ Hermitian matrices $A$ and $B$ the function $f_{A,B}(t)=trace{\, } e^{tA+B}$ is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely "matrix" proof of the BMV conjecture for the special case $rank\,A=1$. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.

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arXiv2016

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.

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arXiv2016

On the roots of a hyperbolic polynomial pencil

Let $ν_0(t),ν_1(t),\,\ldots\,,ν_n(t)$ be the roots of the equation $R(z)=t$, where $R(z)$ is a rational function of the form \[R(z)=z+\sum\limits_{k=1}^n\frac{α_k}{z-μ_k},\] $μ_k$ are pairwise different real numbers, $α_k>0,\,1\leq{}k\leq{}n$. Then for each real $ξ$, the function $e^{ξν_0(t)}+e^{ξν_1(t)}+\,\cdots\,+e^{ξν_n(t)}$ is exponentially convex on the interval $-\infty<t<\infty$.

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arXiv2016

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.

math.FA↗
arXiv2016

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 $σ$ on the unit circle $\mathbb{T}$ generates the scalar product \[\langle\,f\,,\,g\,\rangle_{\!_{L^2_σ}} =\int\limits_{\mathbb{T}}f(t)\,\bar{g(t)}\,σ(dt), \quad \forall\,f,g\,\in\,L^2_σ.\] The measures $σ$ are described which satisfy the condition \[\langle\,f\,,\,g\,\rangle_{\!_{L^2_σ}}= \langle\,f\,,\,g\,\rangle_{\!_{L^2_m}},\quad \forall\,f,g\in\mathscr{L}(z_1,z_2,\,\ldots\,z_n),\] where $m$ is the normalized Lebesgue measure on $\mathbb{T}$.

math.CV↗
arXiv2015

On the BMV conjecture for 2\times2 matrices and the exponential convexity of the function \cosh(\sqrt{at^2+b})

The BMV conjecture states that for \(n\times n\) Hermitian matrices \(A\) and \(B\) the function \(f_{A,B}(t)=\tr e^{tA+B}\) is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely "matrix" proof of the BMV conjecture for \(2\times2\) matrices. This proof is based on the Lie product formula for the exponential of the sum of two matrices. The proof also uses the commutation relations for the Pauli matrices and does not use anything else.

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arXiv2012

Eigenfunctions of the Cosine and Sine Transforms

A description of eigensubspaces of the cosine and sine operators is presented. The spectrum of each of these two operator consists of two eigenvalues (1,\,-1) and their eigensubspaces are infinite--dimensional. There are many possible bases for these subspaces, but most popular are bases constructed from the Hermite functions. We present other "bases" which are not discrete orthogonal sequences of vectors, but continuous orthogonal chains of vectors. Our work can be considered a continuation and further development of results in \textit{Self-reciprocal functions} by Hardy and Titchmarsh: Quarterly Journ. of Math. (Oxford Ser.) \textbf{1} (1930).

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arXiv2011

Stieltjes Functions and Hurwitz Stable Entire Functions

The concept of stability, originally introduced for polynomials, will be extended to apply to the class of entire functions. This generalization will be called Hurwitz stablility and the class of Hurwitz stable functions will serve as the main focus of this paper. A first theorem will show how, given a function of either of the Stieltjes classes, a Hurwitz stable function might be constructed. A second approach to constructing Hurwitz stable functions, based on using additional functions from the Laguerre-Pólya class, will be presented in a second theorem.

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arXiv2009

The Truncated Fourier Operator. V

The Fourier operator truncated on a finite symmetric interval is considered. The limiting behavior of its spectrum is discussed as the length of the interval tends to infinity.

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arXiv2009

The truncated Fourier operator. III

The spectral theory of the Fourier operator (non-truncated) is expounded. The known construction of basis of eigenvectors consisting of the Hermite functions is presented. The detail description of the eigenspaces in the spirit of a work by Hardy and Titchmarsh is done.

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