On a Durrmeyer-type modification of the Exponential sampling series
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On a Durrmeyer‑type modification of the Exponential sampling series Carlo Bardaro1 · Ilaria Mantellini1 Received: 5 August 2020 / Accepted: 3 September 2020 © The Author(s) 2020
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. Keywords Exponential sampling Durrmeyer series · Mellin derivatives · Moments · Voronovskaja formula · Modulus of continuity Mathematics Subject Classification 42C15 · 46E22 · 94A20
1 Introduction The theory of the exponential sampling series of (real or complex)-valued functions f defined over the positive real axis, is a powerful tool for investigating certain phenomena in optical physics, as for example, light scattering, Fraunhofer diffraction etc, (see e.g. [16, 19, 26, 29]). From a mathematical point of view these series were rigorously studied in [18], (see also [4]). The suitable frame for studying these operators is the Mellin analysis, in particular the Mellin transform theory (see [17, 28]). Indeed, the exponential sampling operator represents the counterpart of the classical Shannon sampling series of Fourier analysis (see [27]) in Mellin setting. Now the samples are not equally spaced, but exponentially spaced over the positive real axis, and the classical “sinc” kernel is now replaced by a composition of the sinc-function with the logarithm. The exponential sampling series Carlo Bardaro and Ilaria Mantellini have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica e Applicazioni (GNAMPA)” of the “Instituto di Alta Matematica (INDAM)” as well as by the project “Ricerca di Base 2019 of University of Perugia (title: Integrazione, Approssimazione Analisi non Lineare e loro Applicazioni)”. * Carlo Bardaro [email protected] Ilaria Mantellini [email protected] 1
Department of Mathematics and Computer Sciences, University of Perugia, via Vanvitelli 1, 06123 Perugia, Italy
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enables one to reconstruct functions (signals) which are Mellin-bandlimited. Indeed, the study of the structure of the class of Mellin-bandlimited functions is a deep topic of Mellin analysis. It was studied in [5, 6], in terms of Mellin–Bernstein spaces. As for the Shannon sampling series, later on a generalized version of the exponential sampling series was introduced in [9] (see also [3, 15]), in which the “sinc-log” kernel is replaced by a function 𝜑 defined on ℝ+ satisfying suitable assumptions. This is very important in order to obtain reconstructions of functions not necessarily Mellin band-limited, and to develop a “prediction” theory as a counterpart of the theory developed in [8] for the generalized sampling series of Fourier analysis. In [7] the classical generalized sampling series of Fourier analysis was modified by replacing the sample v
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