Inequalities involving a measure of Marcellán class and zeros of corresponding orthogonal polynomials
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913384447868928 |
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| author | Kumar, Vikash Swaminathan, A. |
| author_facet | Kumar, Vikash Swaminathan, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16014 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inequalities involving a measure of Marcellán class and zeros of corresponding orthogonal polynomials Kumar, Vikash Swaminathan, A. Classical Analysis and ODEs 42C05, 46E22 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. |
| title | Inequalities involving a measure of Marcellán class and zeros of corresponding orthogonal polynomials |
| topic | Classical Analysis and ODEs 42C05, 46E22 |
| url | https://arxiv.org/abs/2308.16014 |