Sampling of real multivariate polynomials and pluripotential theory
Sampling of real multivariate polynomials and pluripotential theory
复制标题
实多元多项式的采样和多能理论
DOI：
10.1353/ajm.2018.0019
复制
发表时间：
2015
影响因子：
1.7
通讯作者：
J. Ortega
中科院分区：
文献类型：
--
作者：
R. Berman;J. Ortega
翻译后摘要：我们考虑的问题，稳定的抽样多变量真实的多项式的大程度的一般框架中的多项式被定义在仿射真实的代数簇M$，配备了加权措施。特别是，这个框架包含了众所周知的三角多项式的设置（当$M$是一个环面配备其不变测度），其中的大次数极限对应于一个高频率极限，以及经典的设置一元正交代数多项式（当$M$是配备有合适的测度的真实的线时），其中采样节点可以被看作是相应正交多项式的零点的推广。它示出的采样的必要条件，在一般的设置，是采样点的渐近密度大于相应的加权平衡措施的密度$M$，定义在多能理论。因此，这个结果推广了著名的朗道型结果的采样环面，其中相应的临界密度对应于Nyqvist率，以及经典的结果说，正交多项式的零点成为均匀分布的对数平衡措施，作为程度趋于无穷大。
abstract:We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when $M$ is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when $M$ is the real line equipped with a suitable measure), where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials. It is shown that a necessary condition for sampling, in the general setting, is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of $M$, as defined in pluripotential theory. This result thus generalizes the well-known Landau type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure, as the degree tends to infinity.
多项式的一些估计
DOI：
--
发表时间：
2009
期刊：
Asian-European Journal of Mathematics 2, no.3
影响因子：
--
作者：
本田竜広;M.Miyagi;M.Nishihara;S.Ohgai;吉田守
通讯作者：
吉田守