Maxim S. Derevyagin

A note on the Geronimus transformation and Sobolev orthogonal polynomials

In this note we recast the Geronimus transformation in the framework of polynomials orthogonal with respect to symmetric bilinear forms. We also show that the double Geronimus transformations lead to non-diagonal Sobolev type inner products.

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 JC=UL is a monic generalized Jacobi matrix associated with the function FC(λ)=λF(λ)+1. It turns out that the Christoffel transformation JC of a bounded monic Jacobi matrix J can be unbounded. This phenomenon is shown to be related to the effect of accumulating at ∞ of the poles of the Pade approximants of the function FC although FC is holomorphic at ∞. The case of the UL-factorization of J is considered as well.

An operator approach to multipoint Padé approximations

First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna-Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a convergence theorem for multipoint Pade approximants to Nevanlinna functions is proved.

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.

Convergence of Diagonal Padé Approximants for a Class of Definitizable Functions

Convergence of diagonal Pade approximants is studied for a class of functions which admit the integral representation \( \mathfrak{F}(\lambda ) = r_1 (\lambda )\int_{ - 1}^1 {\frac{{td\sigma (t)}} {{t - \lambda }} + r_2 (\lambda )} \), where δ is a finite nonnegative measure on [−1, 1], r 1, r 2 are real rational functions bounded at ∞, and r 1 is nonnegative for real λ. Sufficient conditions for the convergence of a subsequence of diagonal Pade approximants of \( \mathfrak{F} \) on ℝ \ [−1, 1] are found. Moreover, in the case when r 1 ≡ 1, r 2 ≡ 0 and δ 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 Pade approximants of \( \mathfrak{F} \) in terms of the so-called generalized Jacobi matrix associated with the asymptotic expansion of \( \mathfrak{F} \) at infinity.

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, as well as 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 ~, we compute the spectrum and the numerical range of the underlying linear pencil explicitly.

On the relation between Darboux transformations and polynomial mappings

Abstract Let d μ be a probability measure on [ 0 , + ∞ ) such that its moments are finite. Then the Cauchy–Stieltjes transform S of d μ 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 S ( λ ) ↦ λ S ( 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.

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 Szegos theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.

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

We recast Stahls 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 Pade approximants is equivalent to the definitizability of the corresponding tridiagonal matrix.