arXiv subjects

Kiran Kumar Behera

Publications and source records attributed to Kiran Kumar Behera.

8 recordsLinked to original sources

arXiv2022

Self-inversive polynomial and quasi-orthogonality on the unit circle

In this paper we study quasi-orthogonality on the unit circle based on the structural and orthogonal properties of a class of self-invariant polynomials. We discuss a special case in which these polynomials are represented in terms of the reversed Szegő polynomials of consecutive degrees and illustrate the results using contiguous relations of hypergeometric functions. This work is motivated partly by the fact that recently cases have been made to establish para-orthogonal polynomials as the unit circle analogues of quasi-orthogonal polynomials on the real line so far as spectral properties are concerned. We show that structure wise too there is great analogy when self-inversive polynomials are used to study quasi-orthogonality on the unit circle.

math.FA↗

arXiv2021

Quadratic transformation and matrix biorthogonal polynomials: an $\mathcal{LU}$ factorization approach

The manuscript presents the $LU$ approach to matrix biorthogonal polynomials when all the even ordered entries in the Gram matrix are zero. This arises in case of a quadratic transformation which is briefly discussed. Further, the main diagonal of the Gram matrix is a zero diagonal and we present the theory that follows from this fact. Precisely, we discuss the Christoffel transformation and matrix representations of the kernel polynomials, usually called the ABC Theorem. Finally, we provide an illustration of our results assuming the Gram matrix has Hankel symmetry.

math.FA↗

arXiv2020

A generalized inverse eigenvalue problem and $m$-functions

In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ of matrices arising in the theory of rational interpolation and biorthogonal rational functions. In addition to the reconstruction of the Hermitian matrix $\mathcal{H}_{[0,n]}$ with the entries $b_j's$, characterizations of the rational functions that are components of the prescribed eigenvectors are given. A condition concerning the positive-definiteness of $\mathcal{J}_{[0,n]}$ and which is often an assumption in the direct problem is also isolated. Further, the reconstruction of $\mathcal{H}_{[0,n]}$ is viewed through the inverse of the pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ which involves the concept of $m$-functions.

math.FA↗

arXiv2018

Biorthogonal rational functions of $R_{II}$ type

In this work, a sequence of orthonormal rational functions that is also biorthogonal to another sequence of rational functions arising from recurrence relations of $R_{II}$ type is constructed. The biorthogonality is proved by a procedure which we call Zhedanov method. A particular case of a sequence of orthonormal rational functions having denominators of special form is considered to motivate the general case. The particular case provides a Christoffel type transformation of the generalized eigenvalue problem with a reformulation different from the existing literature.

math.CA↗

arXiv2018

Biorthogonality and para-orthogonality of $R_I$ polynomials

In this paper, a sequence of linear combination of $R_{I}$ type polynomials such that the terms in this sequence have a common zero is constructed. A biorthogonality relation arising from such a sequence is discussed. Besides a sequence of para-orthogonal polynomials by removing the common zero using suitable conditions is obtained. Finally, a case of hypergeometric functions is studied to illustrate the results obtained.

math.CA↗

arXiv2017

Orthogonal polynomials on the real line corresponding to a perturbed chain sequence

In recent years, chain sequences and their perturbations have played a significant role in characterising the orthogonal polynomials both on the real line as well as on the unit circle. In this note, a particular disturbance of the chain sequence related to orthogonal polynomials having their true interval of orthogonality as a subset of $[0,\infty)$ is studied leading to an important consequence related to the kernel polynomials. Such perturbations are shown to be related to transformations of symmetric measures. An illustration using the generalized Laguerre polynomials is also provided.

math.CA↗

arXiv2017

Orthogonal Polynomials related to g-fractions with missing terms

The purpose of the present paper is to investigate some structural and qualitative aspects of two different perturbations of the parameters of $g$-fractions. In this context the concept of \emph{gap} $g$-fractions is introduced. While tail sequences of a continued fraction play a significant role in the first perturbation, Schur fractions are used in the second perturbation of the $g$-parameters that are considered. Illustrations are provided using Gaussian hypergeometric functions. Using a particular gap $g$-fraction, some members of the class of Pick functions are also identified.

math.CA↗

arXiv2016

Orthogonal Polynomials Associated with Complementary Chain Sequences

Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegö polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.

math.CA↗