Searcharxiv⌕ Search

arXiv subjects

Maxim Derevyagin

Publications and source records attributed to Maxim Derevyagin.

32 records · Page 2Linked to original sources
arXiv2015

On 2D discrete Schrödinger operators associated with multiple orthogonal polynomials

A class of cross-shaped difference operators on a two dimensional lattice is introduced. The main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials. In other words, this scheme generalizes the classical connection between Jacobi matrices and orthogonal polynomials to the case of operators on lattices. Furthermore we also show how to obtain 2D discrete Schrödinger operators out of this construction and give a number of explicit examples based on known families of multiple orthogonal polynomials.

math.CA↗
arXiv2014

Spectral theory of the G-symmetric tridiagonal matrices related to Stahl's counterexample

We recast Stahl's counterexample from the point of view of the spectral theory of the underlying non-symmetric Jacobi matrices. In particular, it is shown that these matrices are self-adjoint and non-negative in a Krein space and have empty resolvent sets. In fact, the technique of Darboux transformations (aka commutation methods) on spectra which is used in the present paper allows us to treat the class of all G-non-negative tridiagonal matrices. We also establish a correspondence between this class of matrices and the class of signed measures with one sign change. Finally, it is proved that the absence of the spurious pole at infinity for Padé approximants is equivalent to the definitizability of the corresponding tridiagonal matrix.

math.SP↗
arXiv2014

Multiple Geronimus transformations

We consider multiple Geronimus transformations and show that they lead to discrete (non-diagonal) Sobolev type inner products. Moreover, it is shown that every discrete Sobolev inner product can be obtained as a multiple Geronimus transformation. A connection with Geronimus spectral transformations for matrix orthogonal polynomials is also considered.

math.CA↗
arXiv2013

Bannai-Ito polynomials and dressing chains

Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.

math-ph↗
arXiv2012

On the relation between Darboux transformations and polynomial mappings

Let dμ(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of dμ(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present paper, we consider a matrix interpretation of the unwrapping transformation from S(λ) to λS(λ^2), which is intimately related to the simplest case of polynomial mappings. More precisely, it is shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to the Carlitz problem appears as a shifted Darboux transformation.

math.CA↗
arXiv2011

Szegő's theorem for matrix orthogonal polynomials

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szegő's theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.

math.CA↗
arXiv2011

CMV matrices and little and big -1 Jacobi polynomials

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter which leads to a linear operator pencil. We show that the little and big -1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.

math.CA↗
arXiv2011

Darboux transformations of Jacobi matrices and Padé approximation

Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_c=UL is a monic generalized Jacobi matrix associated with the function F_c(z)=zF(z)+1. It turns out that the Christoffel transformation J_c of a bounded monic Jacobi matrix J can be unbounded. This phenomenon is shown to be related to the effect of accumulating at infinity of the poles of the Padé approximants of the function F_c although F_c is holomorphic at infinity. The case of the UL-factorization of J is considered as well.

math.CA↗
arXiv2011

Inverse problems for periodic generalized Jacobi matrices

Some inverse problems for semi-infinite periodic generalized Jacobi matrices are considered. In particular, a generalization of the Abel criterion is presented. The approach is based on the fact that the solvability of the Pell-Abel equation is equivalent to the existence of a certainly normalized $J$-unitary $2\times 2$-matrix polynomial (the monodromy matrix).

math.CA↗
arXiv2010

The Jacobi matrices approach to Nevanlinna-Pick problems

A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of $\bR_0$-functions gives rise to a linear pencil $H-λJ$, where $H$ and $J$ are Hermitian tridiagonal matrices. First, we show that $J$ is a positive operator. Then it is proved that the corresponding Nevanlinna-Pick problem has a unique solution iff the densely defined symmetric operator $J^{-1/2}HJ^{-1/2}$ is self-adjoint and some criteria for this operator to be self-adjoint are presented. Finally, by means of the operator technique, we obtain that multipoint diagonal Padé approximants to a unique solution $φ$ of the Nevanlinna-Pick problem converge to $φ$ locally uniformly in $\dC\setminus\dR$. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Padé approximation for $\bR_0$-functions.

math.CA↗
arXiv2010

The linear pencil approach to rational interpolation

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Pade approximants at infinity by considering rational interpolants, (bi-)orthogonal rational functions and linear pencils zB-A of two tridiagonal matrices A, B, following Spiridonov and Zhedanov. In the present paper, beside revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Pade approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to infinity, we compute explicitly the spectrum and the numerical range of the underlying linear pencil.

math.CA↗
arXiv2009

Convergence of diagonal Padé approximants for a class of definitizable functions

Convergence of diagonal Padé approximants is studied for a class of functions which admit the integral representation $ {\mathfrak F}(λ)=r_1(λ)\int_{-1}^1\frac{tdσ(t)}{t-λ}+r_2(λ), $ where $σ$ is a finite nonnegative measure on $[-1,1]$, $r_1$, $r_2$ are real rational functions bounded at $\infty$, and $r_1$ is nonnegative for real $λ$. Sufficient conditions for the convergence of a subsequence of diagonal Padé approximants of $ {\mathfrak F}$ on $\dR\setminus[-1,1]$ are found. Moreover, in the case when $r_1\equiv 1$, $r_2\equiv 0$ and $σ$ has a gap $(α,β)$ containing 0, it turns out that this subsequence converges in the gap. The proofs are based on the operator representation of diagonal Padé approximants of $ {\mathfrak F}$ in terms of the so-called generalized Jacobi matrix associated with the asymptotic expansion of $ {\mathfrak F}$ at infinity.

math.CA↗
arXiv2008

Generalized Jacobi operators in Krein spaces

A special class of generalized Jacobi operators which are self-adjoint in Krein spaces is presented. A description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given. In particular, special attention is paid to the periodic generalized Jacobi operators. Finally, the spectral properties of generalized Jacobi operators are applied to prove convergence results for Padé approximants.

math.SP↗