We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Valle-Poussin operators, Ces`aro operators, Abel opera-tors, and Jacks...We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Valle-Poussin operators, Ces`aro operators, Abel opera-tors, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the L q space for 1≤q 【 ∞, and the Fourier partial summation operators and the Valle-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the L q space for 1≤q 【 ∞.展开更多
The article concerns the average onesided widths of the Sobolev and Besov classes and the classes of functions with bounded moduli of smoothness. The weak asymptotic results are obtained for the corresponding quantities.
基金supported by National Natural Science Foundation of China(Grant No. 10871132)Beijing Natural Science Foundation (Grant No. 1102011)Key Programs of Beijing Municipal Education Commission (Grant No. KZ200810028013)
文摘We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Valle-Poussin operators, Ces`aro operators, Abel opera-tors, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the L q space for 1≤q 【 ∞, and the Fourier partial summation operators and the Valle-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the L q space for 1≤q 【 ∞.
基金Supported by the Foundation of Education Department of Yunnan Province (07Z10533)Supported partly by the National Natural Science Foundation of China (10471010)+1 种基金partly by the project "Representation Theory and Related Topics" of the "985 program" of Beijing Normal UniversitySupported by the Science Foundation of Yunnan University (2008YB027)
文摘The article concerns the average onesided widths of the Sobolev and Besov classes and the classes of functions with bounded moduli of smoothness. The weak asymptotic results are obtained for the corresponding quantities.
