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 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 μ, the anisotropic trigonometric polynomial subspaces are order optimal in the Lq metric for 1⩽q<∞, 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 Lq metric for 1⩽q<∞.

[1]  Vitaly Maiorov.  About Widths of Wiener Space in the Lq-Norm , 1996, J. Complex..

[2]  David Lee,et al.  Approximation of linear functionals on a banach space with a Gaussian measure , 1986, J. Complex..

[3]  Heping Wang,et al.  Approximation of functions on the Sobolev space with a Gaussian measure , 2010 .

[4]  Yanwei Zhang,et al.  Approximation of functions on the Sobolev space on the sphere in the average case setting , 2009, J. Complex..

[5]  Ye Peixin,et al.  Probabilistic and average linear widths of Sobolev space with Gaussian measure , 2003 .

[6]  Heping Wang,et al.  Best approximation of functions on the ball on the weighted Sobolev space equipped with a Gaussian measure , 2010, J. Approx. Theory.

[7]  Chen Guanggui,et al.  Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure , 2004 .

[8]  Vitaly Maiorov,et al.  Average n-Widths of the Wiener Space in the Linfinity-Norm , 1993, J. Complex..

[9]  David Lee,et al.  Approximation of linear operators on a Wiener space , 1986 .

[10]  H. Woxniakowski.  Information-Based Complexity , 1988 .

[11]  Cheng Guanggui,et al.  Linear widths of a multivariate function space equipped with a Gaussian measure , 2005 .

[12]  Jakob Creutzig,et al.  Relations between Classical, Average, and Probabilistic Kolmogorov Widths , 2002 .

[13]  Klaus Ritter,et al.  Average-case analysis of numerical problems , 2000, Lecture notes in mathematics.

[14]  Klaus Ritter,et al.  Approximation and optimization on the Wiener space , 1990, J. Complex..

[15]  Yongsheng Sun,et al.  Average Error Bounds of Best Approximation of Continuous Functions on the Wiener Space , 1995, J. Complex..

[16]  V. Maiorov.  Linear Widths of Function Spaces Equipped with the Gaussian Measure , 1994 .

[17]  A. Pinkus.  n-Widths in Approximation Theory , 1985 .

[18]  G. Lorentz,et al.  Constructive approximation : advanced problems , 1996 .

[19]  Yongsheng Sun,et al.  mu-Average n-Widths on the Wiener Space , 1994, J. Complex..

[20]  Ye Peixin,et al.  Probabilistic and Average Linear Widths of Sobolev Space with Gaussian Measure in L\infty-Norm , 2003 .

[21]  G. Wasilkowski,et al.  Probabilistic and Average Linear Widths inL∞-Norm with Respect tor-fold Wiener Measure , 1996 .

[22]  V. N. Temli︠a︡kov.  Approximation of periodic functions , 1993 .

[23]  孙永生.  AVERAGE n-WIDTH OF POINT SET IN HILBERT SPACE , 1992 .