Zeros of linear combinations of orthogonal polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910950843482112 |
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| author | Durán, Antonio J. |
| author_facet | Durán, Antonio J. |
| contents | Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_jp_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\cdots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$ (which do not depend on $n$). We prove that for every positive measure $μ$ there always exists a sequence of orthogonal polynomials with respect to $μ$ such that all the zeros of the polynomial $q_n$ above are real and simple for $n\ge n_0$, where $n_0$ is a positive integer depending on $K$ and the $γ_j$'s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of linear combinations of orthogonal polynomials Durán, Antonio J. Classical Analysis and ODEs Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_jp_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\cdots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$ (which do not depend on $n$). We prove that for every positive measure $μ$ there always exists a sequence of orthogonal polynomials with respect to $μ$ such that all the zeros of the polynomial $q_n$ above are real and simple for $n\ge n_0$, where $n_0$ is a positive integer depending on $K$ and the $γ_j$'s. |
| title | Zeros of linear combinations of orthogonal polynomials |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2505.11956 |