We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Vallée-Poussin operators, Cesàro operators, Abel operators, 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 Lq space for 1 q < ∞, and the Fourier partial summation operators and the Vallée-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the Lq space for 1 q < ∞. Keywordsaverage width, best approximation, approximation by linear operators, Sobolev space, Gaussian measure MSC(2000): 42A10, 41A25, 41A35, 42A61 Citation: Wang H P, Zhang Y W, Zhai X B. Approximation of functions on the Sobolev space with a Gaussian measure.

This paper contains two parts. In the first part, we obtain the relations between the classical and paverage Kolmogorov widths for all p, 0 < p < ∞, which is a generalization of the corresponding results of J. Creutzig given in [J. Creutzig, Relations between classical, average and probabilistic Kolmogorov widths, J. Complexity 18 (2002) 287-303]. In the second part, we investigate the best approximation of functions on the weighted Sobolev space BW r 2,µ (B d ) equipped with a centered Gaussian measure by polynomial subspaces in the L q,µ metric for 1 ≤ q < ∞, where L q,µ , 1 ≤ q < ∞, denotes the weighted L q space of functions on the unit ball B d with respect to the weight (1 − |x| 2 ) µ− 1 2 , µ ≥ 0. The asymptotic orders of the average error estimations are obtained. We find a striking fact that, in the average case setting, the polynomial subspaces are the asymptotically optimal linear subspaces in the L q,µ metric only for 1 ≤ q < 2 + 1/µ, which means that 2 + 1/µ is the critical value and is independent of dimension d.

a b s t r a c tIn this paper, we discuss the best approximation of functions by spherical polynomials and the approximation by Fourier partial summation operators, Vallée-Poussin operators, Cesàro operators, and Abel operators, on the Sobolev space on the sphere with a Gaussian measure, and obtain the average error estimates. We also get the asymptotic values for the average Kolmogorov and linear widths of the Sobolev space on the sphere and show that, in the average case setting, the spherical polynomial subspaces are the asymptotically optimal subspaces in the L q (1 ≤ q < ∞) metric, and Fourier partial summation operators and Vallée-Poussin operators are the asymptotically optimal linear operators and are (modulo a constant) as good as optimal nonlinear operators in the L q (1 ≤ q < ∞) metric.

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In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the Lq space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the Lq space for 2 < q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the Lq space for 1 ≤ q < ∞. Also, in the Lq space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.