Approximation of multivariate periodic functions on the L2 space with a Gaussian measure
Approximation of multivariate periodic functions on the L2 space with a Gaussian measure
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DOI：
10.1016/j.jmaa.2011.10.034
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发表时间：
2012-04
影响因子：
1.3
通讯作者：
Heping Wang;Wei-Sun Jiang;Xuebo Zhai
中科院分区：
文献类型：
--
作者：
Heping Wang;Wei-Sun Jiang;Xuebo Zhai
本文主要研究多元周期函数在平均情形下的逼近问题。我们在多元周期函数的L2空间中引入一个高斯测度μ，使得它的Cameron-Martin空间是各向异性的多元周期空间.在此Gauss测度下，讨论了三角多项式对平行六面体函数的最佳逼近以及相应的各向异性Fourier部分和算子和Vallée-Poussin算子的逼近，并得到了平均误差估计.我们将证明，在平均情况下，平均值是关于这个高斯测度μ的，各向异性三角多项式子空间在Lq度量中是阶最优的，并且各向异性傅里叶部分和算子和Vallée-Poussin算子是阶最优线性算子，它们与在Lq度量中的最优非线性算子一样好。
This paper is devoted to studying the approximation of multivariate periodic functions in the average case setting. We equip the L2space of multivariate periodic functions with a Gaussian measure μ such that its Cameron–Martin space is the anisotropic multivariate periodic space. With respect to this Gaussian measure, we discuss the best approximation of functions by trigonometric polynomials with harmonics from parallelepipeds and the approximation by the corresponding anisotropic Fourier partial summation operators and Vallée-Poussin operators, and get the average error estimation. We shall show that, in the average case setting, with the average being with respect to this Gaussian measure μ, the anisotropic trigonometric polynomial subspaces are order optimal in the Lqmetric for 1⩽q<∞, and the anisotropic Fourier partial summation operators and Vallée-Poussin operators are the order optimal linear operators, which are as good as optimal nonlinear operators in the Lqmetric for 1⩽q<∞.