On a Durrmeyer-type modification of the Exponential sampling series

Abstract

In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.

Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces

“…Let ϕ ∈ Φ and ψ ∈ Ψ. For any f : R + → C, we define the exponential sampling Durrmeyer series of f as (see [13])…”

“…In the recent paper [13] we introduced the so-called Durrmeyer-type exponential sampling operator, following the same idea developed for the generalized sampling series. We replace the sampled-value of a function f by a convolution integral operator of Mellin type (see [17]).…”

Semi‐discrete operators in multivariate setting: Convergence properties and applications

“…They are called (univariate) “exponential sampling series.” The generalized version of such series was introduced in Bardaro et al [11]. Later on Kantorovich [12] versions were studied in previous studies [13–15] while a Durrmeyer version was introduced in Bardaro and Mantellini [16] (see also Bajpeyi et al [17]). In Bardaro et al [18], a two‐dimensional version was studied with the aim to obtain mathematical models for the study of the propagation of seismic waves (a general multivariate version was recently studied in Kursun et al [19]).…”

Approximation of discontinuous functions by Kantorovich exponential sampling series

“…Direct convergence results and the improved order of approximation for Kantorovich exponential sampling series has been analysed in [31]. Further, the Durrmeyer type modification of exponential sampling series (1.1) is considered in [1] and [30]. So far the approximation of continuous functions by Kantorovich exponential sampling series I χ w f has been studied.…”