Lp-Sampling recovery for non-compact subclasses of L∞

In this article, we study the sampling recovery problem for certain relevant multivariate function classes on the cube [0, 1]d, which are not compactly embedded into L∞([0,1]d). Recent tools relating the sampling widths to the Kolmogorov or best m-term trigonometric widths in the uniform norm are therefore not applicable. In a sense, we continue the research on the small smoothness problem by considering limiting smoothness in the context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity such that the sampling recovery problem is still relevant. There is not much information available on the recovery of such functions except for a previous result by Oswald in the univariate case and Dinh Dũng in the multivariate case. As a first step, we prove the uniform boundedness of the ℓp-norm of the Faber coefficients at a fixed level by Fourier analytic means. Using this, we can control the error made by a (Smolyak) truncated Faber series in Lq([0,1]d) with q <∞. It turns out that the main rate of convergence is sharp. Thus, we obtain results also for S1,∞1F([0,1]d), a space “close” to S11W([0,1]d), which is important in numerical analysis, especially numerical integration, but has rather poor Fourier analytical properties.