Lp-Sampling recovery for non-compact subclasses of L∞ - Byrenheid, Glenn; Stasyuk, Serhii; Ullrich, Tino (Frontiers in Applied Mathematics and Statistics, 2023)
In this paper we study the sampling recovery problem for certain relevant multivariate function classes on the cubeRecent 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 still makes sense. There is not much known on the recovery of such functions except of an old result by Oswald in the univariate situation and results by Dinh Dũng for the multivariate case. As a first step we prove the uniform boundedness of the ℓ p -norm of the Faber coefficients in a fixed level by Fourier analytic means. Using this we are able to control the error made by a (Smolyak) truncated Faber series in L q ([0, 1] d ) 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 1 W ([0, 1] d ) which is important in numerical analysis, especially numerical integration, but has rather bad Fourier analytical properties.