Approximation by sampling type discrete operators

Abstract

In this paper, we deal with discrete operators of sampling type. It is known that this type of operators are related to generalized sampling series and they have important applications. In this work, using bounded and uniformly continuous functions we get general estimations under usual supremum norm with the help of summability method. We also study the degree of approximation with respect to suitable Lipschitz class of continuous functions. Finally, we give speci…c kernels which verify our kernel assumptions.

Approximation by pseudo-linear discrete operators

“…Introduction. Discrete operators are a notable research area in approximation theory, and many significant approximation results have been obtained in various forms (see [2,3,4,6,15,16,17,18,19,20,38]). These operators are connected with generalized sampling series.…”

“…which are introduced and studied in [3,4,6], where the discrete kernel actualize the following conditions…”

“…These conditions are known in the literature as approximate identities. One can find or construct many different kernels satisfying the above conditions (see [3,4,6]). We remind that Bede et al in [13], change the structure of and • by ⊕ and respectively and construct pseudo-linear operators with continuous kernels as follows…”

“…For these type of operators, we need some usual conditions given by p 1 ) k∈Z p k,w = 1 for all w > 0, p 2 ) there exists a r > 0 such that lim w→∞ |k|≥r p k,w = 0, which are natural in this setting (see [3,4,6]). Before giving our approximation theorem, we mention about some features of pseudo-operations and pseudo-linear operators with the following lemmas and propositions.…”

“…In this section, we examine the order of convergence by means of g-modulus of continuity. To this end, as in [5,6,7,8,9,10,34], we take into account the following definitions. Let (u w ) and (v w ) be a sequence of positive real numbers.…”