Approximation of multivariate periodic functions on the L-2 space with a Gaussian measure

作者：Wang, Heping^*; Jiang, Weigang; Zhai, Xuebo

来源：Journal of Mathematical Analysis and Applications, 2012, 388(2): 929-941.

DOI：10.1016/j.jmaa.2011.10.034

• Average width
• Best approximation
• Approximation by linear operators
• Anisotropic space
• Gaussian measure

This paper is devoted to studying the approximation of multivariate periodic functions in the average case setting. We equip the L-2 space of multivariate periodic functions with a Gaussian measure mu 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 Vallee-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 mu, the anisotropic trigonometric polynomial subspaces are order optimal in the L-q metric for 1 <= q < infinity, and the anisotropic Fourier partial summation operators and Vallee-Poussin operators are the order optimal linear operators, which are as good as optimal nonlinear operators in the L-q metric for 1 <= q < infinity.

• 出版日期2012-4-15
• 单位首都师范大学