逆问题、索博列夫-切比雪夫多项式和渐近性

IF 0.5 4区 数学 Q3 MATHEMATICS
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引用次数: 0

摘要

设 (u, v) 是一对准无限对称线性函数,{Pn}n≥0 和 {Qn}n≥0 分别是单正交多项式(SMOP)序列。我们对一元多项式序列 {Rn}n≥0 的定义如下: \(\begin{array}{cc}\frac{{P}_{n+2}^{\mathrm{^{\prime}}}\left(x\right)}{n+2}+{b}_{n}\frac{{P}_{n}^{\mathrm{^{\prime}}}\left(x\right)}{n}-{Q}_{n+1}\left(x\right)={d}_{n-1}\left(x\right),& n\ge 1.\end{array}\) 我们提出了{Rn}n≥0 相对于准无限线性函数 w 正交的必要条件和充分条件。此外,我们还考虑了{Pn}n≥0和{Qn}n≥0分别是第一种和第二种单切比雪夫多项式的情况,并研究了关于索波列夫内积 \(\langle p.)正交的索波列夫多项式的相对外渐近线、q\rangle s=\underset{-1}{\overset{1}{\int }}pq{\left(1-{x}^{2}\right)}^{-1/2}dx+{\uplambda }_{1}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}}{\left(1-{x}^{2}\right)}^{1/2}dx+{\uplambda }_{2}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}d\mu \left(x\right),\),其中 μ 是与 w 相关联的正 Borel 度量,λ1, λ2 >;0; λ2 是 λ1 的线性多项式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics

Let (u, v) be a pair of quasidefinite and symmetric linear functionals with {Pn}n≥0 and {Qn}n≥0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {Rn}n≥0 as follows:

\(\begin{array}{cc}\frac{{P}_{n+2}^{\mathrm{^{\prime}}}\left(x\right)}{n+2}+{b}_{n}\frac{{P}_{n}^{\mathrm{^{\prime}}}\left(x\right)}{n}-{Q}_{n+1}\left(x\right)={d}_{n-1}\left(x\right),& n\ge 1.\end{array}\)

We present necessary and sufficient conditions for {Rn}n≥0 to be orthogonal with respect to a quasidefinite linear functional w. In addition, we consider the case where {Pn}n≥0 and {Qn}n≥0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product

\(\langle p,q\rangle s=\underset{-1}{\overset{1}{\int }}pq{\left(1-{x}^{2}\right)}^{-1/2}dx+{\uplambda }_{1}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}}{\left(1-{x}^{2}\right)}^{1/2}dx+{\uplambda }_{2}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}d\mu \left(x\right),\)

where μ is a positive Borel measure associated with w and λ1, λ2 > 0; λ2 is a linear polynomial of λ1.

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来源期刊
Ukrainian Mathematical Journal
Ukrainian Mathematical Journal MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
0.90
自引率
20.00%
发文量
107
审稿时长
4-8 weeks
期刊介绍: Ukrainian Mathematical Journal publishes articles and brief communications on various areas of pure and applied mathematics and contains sections devoted to scientific information, bibliography, and reviews of current problems. It features contributions from researchers from the Ukrainian Mathematics Institute, the major scientific centers of the Ukraine and other countries. Ukrainian Mathematical Journal is a translation of the peer-reviewed journal Ukrains’kyi Matematychnyi Zhurnal, a publication of the Institute of Mathematics of the National Academy of Sciences of Ukraine.
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