双正交多项式与零映射变换
Biorthogonal polynomials and zero-mapping transformations
摘要：
作者在\cite{IN2}中提出了一种生成$n$次多项式的集合${\Bbb P}_n$到其自身的变换$\cal T$的技术，使得如果$p\in{\Bbb P}_n$在$(c,d)$中具有其所有的零，则${\cal T}\{p\}$在$(a,b)$中具有其所有的零，其中$(a,b)$和$(c,d)$被给定实区间。该技术依赖于推导的一个明确的形式的双正交多项式的Borel措施是严格的符号一致，使连续的广义矩的比率是一个合理的$[1/1]$函数的参数。\cite{IN2}中讨论过的严格签署一致措施的具体实例包括$x^μ\Dψ(x)$、$μ^x\Dψ(x)$和$x^{\log_qμ}\Dψ(x)$、$q\in(0,1)$。在本文中，我们确定所有措施$ψ$，使其连续的广义矩有一个合理的$[1/1]$商，从而表征所有可能的零映射变换的这种。
英文摘要：
The authors have presented in \cite{IN2} a technique to generate transformations $\cal T$ of the set ${\Bbb P}_n$ of $n$th degree polynomials to itself such that if $p\in{\Bbb P}_n$ has all its zeros in $(c,d)$ then ${\cal T}\{p\}$ has all its zeros in $(a,b)$, where $(a,b)$ and $(c,d)$ are given real intervals. The technique rests upon the derivation of an explicit form of biorthogonal polynomials whose Borel measure is strictly sign consistent and such that the ratio of consecutive generalized moments is a rational $[1/1]$ function of the parameter. Specific instances of strictly sign consistent measures that have been debated in \cite{IN2} include $x^μ\Dψ(x)$, $μ^x\Dψ(x)$ and $x^{\log_qμ}\Dψ(x)$, $q\in(0,1)$. In this paper we identify all measures $ψ$ such that their consecutive generalized moments have a rational $[1/1]$ quotient, thereby characterizing all possible zero-mapping transformations of this kind.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
