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arXiv · 2308.16014

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

Vikash Kumar·A. Swaminathan
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Abstract

Let $\tildeΦ_n$ 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 $\tildeμ$ such that $\tildeΦ_n$ 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 $\tildeΦ_n$. 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 $\tildeΦ_n$ are obtained by involving their measures. This leads to the study of the Lubinsky type inequality between the measures $μ$ and $\tildeμ$, without using the ordering relation between $μ$ and $\tildeμ$. Additionally, similar type of inequalities for the kernel type polynomials related to $μ$ and $\tildeμ$ are obtained.

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Vikash Kumar, A. Swaminathan. 2024-06-11. Inequalities involving a measure of Marcellán class and zeros of corresponding orthogonal polynomials. https://arxiv.org/abs/2308.16014

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