L_p-Sampling recovery for non-compact subclasses of L_∞
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In this paper we study the sampling recovery problem for certain relevant multivariate function classes which are not compactly embedded into L_∞. Recent tools relating the sampling numbers to the Kolmogorov widths in the uniform norm are therefore not applicable. In a sense, we continue the research on the small smoothness problem by considering "very" small smoothness in the context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity. There is not much known on the recovery of such functions except of an old result by Oswald in the univariate situation. As a first step we prove the uniform boundedness of the ℓ_p-norm of the Faber-Schauder coefficients in a fixed level. Using this we are able to control the error made by a (Smolyak) truncated Faber-Schauder series in L_q with q<∞. It turns out that the main rate of convergence is sharp. As a consequence we obtain results also for S^1_1,∞F([0,1]^d), a space which is “close” to the space S^1_1W([0,1]^d) which is important in numerical analysis, especially numerical integration, but has rather bad Fourier analytic properties.
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