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Mediterr. J. Math. (2023) 20:263
https://doi.org/10.1007/s00009-023-02459-2
1660-5446/23/050001-21
published online July 10, 2023
c
The Author(s), under exclusive licence to Springer
Nature Switzerland AG 2023
Approximation Results for Hadamard-Type
Exponential Sampling Kantorovich Series
Sadettin Kursun, Ali Aral and Tuncer Acar
Abstract. The present paper deals with construction of a new
family of exponential sampling Kantorovich operators based on a suit-
able fractional-type integral operators. We study convergence properties
of newly constructed operators and give a quantitative form of the rate
of convergence thanks to logarithmic modulus of continuity. To obtain
an asymptotic formula in the sense of Voronovskaja, we consider locally
regular functions. The rest of the paper devoted to approximations of
newly constructed operators in logarithmic weighted space of functions.
By utilizing a suitable weighted logarithmic modulus of continuity, we
obtain a rate of convergence and give a quantitative form of Voronovskaja-
type theorem via remainder of Mellin–Taylor’s formula. Furthermore,
some examples of kernels which satisfy certain assumptions are pre-
sented and the results are examined by illustrative numerical tables and
graphical representations.
Mathematics Subject Classification. 41A35, 30D10, 94A20, 41A25, 26A33,
44A15.
Keywords. Exponential sampling Kantorovich series, Hadamard-type
fractional integral operators, rate of convergence, modulus of continuity,
logarithmic weighted space of functions, Voronovskaja-type formulae.
1. Introduction
The generalized sampling-type operators were developed by E. T. Whittaker
[58], V. A. Kotel’nikov [49] and C. E. Shannon [56]. The theory and applica-
tions of sampling-type operators are among the most challenging areas in the
field of approximation theory, especially in the domains of image and signal
processing (see [13,27]). The family of operators is given by
Ssinc
wg(˜x):=
k∈Z
gk
wsinc (w˜x−k),˜x∈R,w > 0,(1.1)
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