Function Approximation On Sobolev Space With Gaussian Measure

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by the Fourier partial summation operators, the ValleePoussinoperators, the Cesáro operators, the Abel operators, and the 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 asymptotic optimal subspaces in L_q space for 1≤q <∞, and the Fourier partial summation operators and the Vallee-Poussin operators are the asymptotic optimal linear operators and are as good as optimal nonlinear operators in L_q space for 1≤q <∞. Also, the Cesáro operators, the Abel operators and the Jackson operators have the saturation property.