Quasi-Natured Spectral Polynomials: Orthogonality, Zeros and Specific Norm Type Inequalities

Title:       Quasi-Natured Spectral Polynomials: Orthogonality, Zeros and Specific Norm Type Inequalities

Abstract:  In this presentation, I will discuss the various properties of the linear combinations of two consecutive elements of polynomials generated by spectral transformations. These are referred to as “quasi-spectral polynomials” of order one with respect to a given linear spectral transformation. When a linear combination of two consecutive spectral polynomials is formed, the resulting polynomials may not retain their orthogonality. One of the key objective is to address the challenge of restoring orthogonality through what we refer to as “quasi-spectral polynomial” of order one. On the other hand, we study the behavior of the zeros with reference to the support of the measure for specific quasi- spectral polynomials. Further, for the unit circle case, I will discuss various properties related to the M2 class. Along with numerical examples, we analyze the zeros of orthogonal polynomials related to the M2 class, corresponding para-orthogonal polynomial and its linear combination. Comparison of the norm inequalities among the orthogonal polynomials related to M2 class are obtained

by involving their measures. This leads to the study of the Lubinsky type inequality for M2 class without using the assumption of ordering relation between the measures.

Keywords: Orthogonal Polynomials; Quasi-Orthogonal Polynomials; Linear Spectral transformation; Chrsitoffel-Darboux Kernel; Para-orthogonal polynomials on the unit circle; Stieltjes transform; Reproducing Kernel.

2020 AMS Subject Classification:42C05, 33C45, 33C05, 46E22.

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