Semi-discrete moduli of smoothness and their applications in one- and two- sided error estimates

Abstract: In this paper, we introduce a new semi-discrete modulus of smoothness, which generalizes the definition given by Kolomoitsev and Lomako (KL) in 2023 (in the paper published in the J. Approx. Theory), and we establish very general one- and two- sided error estimates under non-restrictive assumptions. The proposed results have been proved exploiting the regularization and approximation properties of certain Steklov integrals introduced by Sendov and Popov in 1983, and differ from the ones given by Kolomoitsev and Lomako. In addition, the proof of the original KL approximation theorems were strictly related to the application of certain classical results of the trigonometric best approximation, and thus, they are applicable only for operators of the trigonometric type. By the definition of semi-discrete moduli of smoothness here proposed, we are able to deduce applications also for operators that are not necessarily of the trigonometric type, and can also be used to derive sharper estimates than those that can be achieved by the classical averaged moduli of smoothness ($\tau$-moduli). Furthermore, a Rathore-type theorem is established, and a new notion of K-functional is also introduced showing its equivalence with the semi-discrete modulus of smoothness and its realization. One-sided estimates of approximation can be established for classical operators on bounded domains, such as the Bernstein polynomials. In the case of approximation operators on the whole real line, one-sided estimates can be achieved, e.g., for the Shannon sampling (cardinal) series, as well as for the so-called generalized sampling operators. At the end of the paper, the case of algebraic Lagrange approximation has been considered, showing the main open problems in order to derive two-sided error estimates in this noteworthy case.