超球多项式的下包络
The lower envelope of ultraspherical polynomials
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中文总结 AI 辅助
本文证明了归一化超球多项式在相邻最大零点区间内的逐点下包络等于对应阶多项式，并解答了 Oliveira Filho 2009 年论文中的问题 2。
中文摘要 AI 辅助
设 $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ 为归一化雅可比多项式。对于 $\alpha\ge0$，令 $\xi_0=-1$，并令 $\xi_n$ 为 $R_n^{(\alpha+1,\alpha)}$ 的最大零点（$n\ge1$）。我们证明：对每个 $k\ge1$，有 $R_k^{(\alpha,\alpha)}(x)=\min_{n\ge0}R_n^{(\alpha,\alpha)}(x)$，其中 $\xi_{k-1}\le x\le\xi_k$。这确定了归一化超球多项式的逐点下包络，并回答了 F. M. de Oliveira Filho 在 2009 年论文中提出的问题 2。
英文摘要
Let $R_n^{(a,b)}=P_n^{(a,b)}/P_n^{(a,b)}(1)$ be the normalised Jacobi polynomials. For $α\ge0$, put $ξ_0=-1$ and let $ξ_n$ be the largest zero of $R_n^{(α+1,α)}$ for $n\ge1$. We prove that $$ R_k^{(α,α)}(x) =\min_{n\ge0}R_n^{(α,α)}(x), \quad ξ_{k-1}\le x\leξ_k, $$ for every $k\ge1$. This determines the pointwise lower envelope of the normalised ultraspherical polynomials and answers Question 2 posed by F. M. de Oliveira Filho in his 2009 thesis. For $α>0$ and $-1<x<1$, we also determine all minimising degrees. The sequence is non-increasing up to each such degree, which answers his Question 1 in this range. A Legendre example gives a negative answer when $α=0$.
发表机构
- University of Coimbra（科英布拉大学）
- Institute of Mathematics, The Ministry of Science and Education（阿塞拜疆科学与教育部数学研究所）
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