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Approximation of discontinuous signals by exponential‐type generalized sampling Kantorovich series

July 2024
Mathematical Methods in the Applied Sciences 48(1):340-355

DOI:10.1002/mma.10330

Authors:

Sadettin Kursun

Tuncer Acar

Selçuk University

In the present paper, we analyze the behavior of the exponential‐type generalized sampling Kantorovich operators Kωφ,G

{K}_{\omega}^{\varphi, \mathcal{G}}

when discontinuous signals are considered. We present a proposition for the series Kωφ,G

{K}_{\omega}^{\varphi, \mathcal{G}}

, and we prove using this proposition certain approximation theorems for discontinuous functions. Furthermore, we give several examples of kernels satisfying the assumptions of the present theory. Finally, some numerical computations are performed to verify the approximation of discontinuous functions f f by Kωφ,Gf

{K}_{\omega}^{\varphi, \mathcal{G}}f

Graph of third‐order Mellin–spline kernel. [Colour figure can be viewed at wileyonlinelibrary.com]

…

Graph of Mellin–Jackson kernel for c=0,n=1,γ=1$$ c&amp;#x0003D;0,n&amp;#x0003D;1,\gamma &amp;#x0003D;1 $$. [Colour figure can be viewed at wileyonlinelibrary.com]

…

Graph of the kernel B˜3$$ {\tilde{B}}_3 $$. [Colour figure can be viewed at wileyonlinelibrary.com]

…

Graph of the kernel B3$$ {\mathcal{B}}_3 $$. [Colour figure can be viewed at wileyonlinelibrary.com]

…

Graph of the function f$$ f $$. [Colour figure can be viewed at wileyonlinelibrary.com]

…

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Received: 15 March 2024 Revised: 24 June 2024 Accepted: 26 June 2024

DOI: 10.1002/mma.10330

RESEARCH ARTICLE

Approximation of discontinuous signals by

exponential-type generalized sampling Kantorovich series

Sadettin Kursun1Tuncer Acar2

1Turkish Military Academy, Department

of Basic Sciences, National Defence

University, Ankara, Turkey

2Faculty of Science, Department of

Mathematics, Selcuk University, Konya,

Turkey

Correspondence

Sadettin Kursun, Turkish Military

Academy, Department of Basic Sciences,

National Defence University, Çankaya,

06420, Ankara, Turkey.

Email: sadettinkursun@yahoo.com

Communicated by: M. A. Ragusa

Funding information

There are no funders to report for this

submission.

In the present paper, we analyze the behavior of the exponential-type gener-

alized sampling Kantorovich operators K𝜑,

𝜔when discontinuous signals are

considered. We present a proposition for the series K𝜑,

𝜔, and we prove using

this proposition certain approximation theorems for discontinuous functions.

Furthermore, we give several examples of kernels satisfying the assumptions

of the present theory. Finally, some numerical computations are performed to

verify the approximation of discontinuous functions 𝑓by K𝜑,

𝜔𝑓.

KEYWORDS

discontinuous functions, exponential-type generalized sampling Kantorovich series,

jump/removable discontinuities, special kernels

MSC CLASSIFICATION

41A35, 41A25, 26A15

1INTRODUCTION

The classical sampling type operators are approximate version of the Whittaker–Kotel'nikov–Shannon (WKS) sampling

theorem developed by Whittaker [1], Kotel'nikov [2], and Shannon [3]. The family of operators is defined by

G𝜔,sinc 𝑓(x)∶=

k∈Z

𝑓k

𝜔sinc (𝜔x−k),x∈R,𝜔 > 0,(1.1)

where sinc function is given by sinc (x)= sin (𝜋x)

𝜋xif x∈R∖{0}and sinc (0)=1.

In 1987, Butzer et al. [4] reconstructed the series (1.1) using a suitable kernel, denoted by 𝜒, which satisfies the typical

assumptions of approximate identities instead of sinc function. The reconstructed series is of the form

G𝜒

𝜔𝑓(x)∶=

k∈Z

𝑓k

𝜔𝜒(𝜔x−k)(1.2)

for any bonded function 𝑓∶R→R.

It is very well-known that Kantorovich's idea allows to approximate not necessarily continuous function. Kantorovich's

idea is based on using integral mean values on the interval k

𝜔,k+1

𝜔of the corresponding function 𝑓instead of its samples

values 𝑓k

𝜔. The Kantorovich form of (1.2) is defined by

K𝜒

𝜔𝑓(x)∶=

k∈Z

𝜒(𝜔x−k)𝜔k+1

𝜔

𝑓(𝜈)d𝜈,(1.3)

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