Smoothness of functions versus smoothness of approximation processes

Abstract

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.

Sharp $L_p$-error estimates for sampling operators

“…Proof. We follow the proof of [11, Lemma 8], see also [18]. Applying (4.2) and using the condition X n ⊂ X 2n , we get…”

“…This holds, for example, for polynomials of near best approximation, de la Vallée Poussin means, corresponding Riesz means, etc. For various applications of realizations of the K-functionals see e.g., [9], [18]- [20]. Below, we give an analogue of equivalence (4.26) for the sampling operator G n .…”

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

“…Following similar arguments as those in [30], let us prove the left-hand side inequality in (1.10). Using Hardy's inequality (2.3)…”

“…Recently [30], smoothness properties of approximation processes were used to characterize smoothness properties of functions themselves. We continue this line of research in L p with Dunkl weights.…”

“…3.5]. Another characterization of Besov spaces via smoothness of approximation processes has been suggested in [30].…”