在单位圆盘上用正交多项式投影的Sobolev范数测量的误差
Orthogonal polynomial projection error measured in Sobolev norms in the unit disk
AI总结：
研究了在单位圆盘上以广义切比雪夫型权重的加权L²正交投影的逼近性质，测量在Sobolev范数中，利用Zernike多项式展开分析，证明了主要结果的准确性并辅以数值测试。
摘要：
我们研究了在单位圆盘上，以广义切比雪夫型权重(1-|x|²)^α的加权L²正交投影到次数小于等于N的多项式空间的逼近性质，这些性质在Sobolev型范数中测量，所有范数均在同一加权L²范数中。我们的基本工具是Zernike多项式展开的分析。在某些情况下证明了主要结果的准确性，其他情况下则通过报告的数值测试强烈暗示。获得了一些具有独立兴趣的辅助结果，包括关于均匀和非均匀加权Sobolev空间的性质，一个Markov型不等式，Zernike多项式之间的连接系数，以及函数及其导数的Fourier-Zernike展开之间的关系。
英文摘要：
We study approximation properties of weighted $L^2$-orthogonal projectors onto the space of polynomials of degree less than or equal to $N$ on the unit disk where the weight is of the generalized Gegenbauer form $x \mapsto (1-|x|^2)^α$. The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted $L^2$ norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases and otherwise strongly hinted at by reported numerical tests. A number of auxiliary results of independent interest are obtained including some properties of the uniformly and non-uniformly weighted Sobolev spaces involved, a Markov-type inequality, connection coefficients between Zernike polynomials and relations between the Fourier-Zernike expansions of a function and its derivatives.
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。
