Orthogonal polynomials and their derivatives, I
Orthogonal polynomials and their derivatives, I
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正交多项式及其导数，I
DOI：
10.1016/0021-9045(84)90023-6
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发表时间：
1984
影响因子：
0.9
通讯作者：
P. Nevai
中科院分区：
文献类型：
--
作者：
S. Bonan;P. Nevai
自1915年Luzin[16，第501页]提出除了三角系统外，是否存在在微分或积分下不变的正交系统以来，已经进行了几项研究，以寻找导数满足某些条件的所有正交多项式。这些问题已经得到了解决，例如[2,4 - 13,15,19,201]。本文给出了导数为同一系统中至多两个多项式的线性组合的所有正交多项式的完备刻划。设a为实线上具有无限支撑力和有限力矩的有限正测度。这样的测度da称为一个分布，对应的标准正交多项式系统表示为(p,], “…其中p，(x)= p,(da, x)= y,(da) x ” +..A, yn, b>0。这些多项式p满足三项递归关系
Ever since 19 15 when Luzin [16, p. 501 asked whether there are any orthogonal systems in addition to the trigonometric system that are invariant under either differentiation or integration there have been several investigations conducted towards finding all the orthogonal polynomials whose derivatives satisfy certain conditions. Such problems have been solved, for example, in [2, 4-13, 15, 19, 201. In this paper we give a complete characterization of all orthogonal polynomials whose derivatives are linear combinations of at most two polynomials of the same system.Let da be a finite positive measure on the real line with infinite support and finite moments. Such a measure da will be called a distribution and the corresponding system of orthonormal polynomials is denoted by (p,],“,,, where p,(x)= p,(da, x)= y,(da) x”+.. a, yn> 0. These polynomials p,, satisfy the three-term recurrence relation