Inequalities involving a measure of Marcellán class and zeros of corresponding orthogonal polynomials

2023/08/30 by Kumar, Vikash, Swaminathan, A.
#42C05 #46E22 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.16014

Abstract

Let Φ_n be a quasi-orthogonal polynomial of order 1 on the unit circle, obtained from an orthogonal polynomial Φ_n with measure μ, which is in the Marcellán class, if there exist another measure μ such that Φ_n is a monic orthogonal polynomial. This article aims to investigate various properties related to the Marcellán class. At first, we study the behaviour of the zeros between Φ_n and Φ_n. Along with numerical examples, we analyze the zeros of Φ_n, its POPUC and the linear combination of the POPUC. Further, comparison of the norm inequalities among Φ_n and Φ_n are obtained by involving their measures. This leads to the study of the Lubinsky type inequality between the measures μ and μ, without using the ordering relation between μ and μ. Additionally, similar type of inequalities for the kernel type polynomials related to μ and μ are obtained.

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