Approximation of multivariate periodic functions on the L2 space with a Gaussian measure

This paper is devoted to studying the approximation of multivariate periodic functions in the average case setting. We equip the L2 space 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 Lq metric 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 Lq metric for 1⩽q<∞.