Abstract
We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Vallée-Poussin operators, Ces`aro 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 L q 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 L q space for 1 ⩽ q < ∞.
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Wang, H., Zhang, Y. & Zhai, X. Approximation of functions on the Sobolev space with a Gaussian measure. Sci. China Math. 53, 373–384 (2010). https://doi.org/10.1007/s11425-009-0113-8
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DOI: https://doi.org/10.1007/s11425-009-0113-8