A Quantitative Estimate for the Sampling Kantorovich Series in Terms of the Modulus of Continuity in Orlicz Spaces

Abstract

In the present paper we establish a quantitative estimate for the sampling Kantorovich operators with respect to the modulus of continuity in Orlicz spaces defined in terms of the modular functional. At the end of the paper, concrete examples are discussed, both for what concerns the kernels of the above operators, as well as for some concrete instances of Orlicz spaces.

Quantitative estimates for nonlinear sampling Kantorovich operators

“…In the particular case of L papproximation, we directly established a quantitative estimate for the order of approximation, with the main purpose to obtain a sharper estimate than that one achieved in the general case. If the latter estimated is applied for the linear version of the sampling Kantorovich operators, we become able to improve the result that could be derived from Theorem 3.1 of [26]. Finally, we give some concrete examples of nonlinear sampling Kantorovich operators constructed by using Fejér and Bspline kernels, establishing some particular results in these instances.…”

“…Concerning the problem of the order of approximation for the (linear) sampling Kantorovich operators, a quantitative estimate in the setting of Orlicz spaces in terms of modulus of continuity has been very recently established in [26]. On the other hand, quantitative estimates with respect to the Jordan variation for sampling-type operators have been obtained in [2] exploiting a suitable modulus of smoothness for the space of absolutely continuous functions AC (R).…”

Blending‐type approximation by Lupaş–Durrmeyer‐type operators involving Pólya distribution

“…Srivastava et al 28 obtained approximation properties on Banach space with the help of deferred Nörlund statistical convergence (see also previous studies 29–32 ). Some amusing approximation properties can be seen in previous studies 33–45 and references therein.…”

Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces

“…In [3], Faried and Bakery studied the operator ideals constructed by − numbers, generalized Cesáro and Orlicz sequence spaces ℓ and show that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions. Also summation process and sequences spaces applications are closely related to Korovkin type approximation theorems and linear positive operators studied by Costarelli and Vinti [4] and Altomare [5]. The idea of this paper is to examine a generalized class by using Orlicz-Cesáro mean sequence spaces and the sequence of -numbers, for which constructs an operator ideal.…”