ArticlePublisher preview available

Approximation by Multivariate Max-Product Kantorovich Exponential Sampling Operators

Authors:
To read the full-text of this research, you can request a copy directly from the author.

Abstract

The approximation behavior of multivariate max-product Kantorovich exponential sampling operators has been analyzed. The point-wise and uniform approximation theorem for these sampling series Iw,jχ,(M)Iw,jχ,(M)I^{\chi ,(M)}_{\textbf{w},j} is proved. The degree of approximation in-terms of logarithmic modulus of smoothness is studied. For the class of log-Hölderian functions, the order of uniform norm convergence is established. The norm-convergence theorems for the multivariate max-product Kantorovich exponential sampling operators in Mellin–Lebesgue spaces is studied.
Results Math (2024) 79:66
c
2024 The Author(s), under exclusive licence to
Springer Nature Switzerland AG
1422-6383/24/020001-22
published online January 13, 2024
https://doi.org/10.1007/s00025-023-02092-1 Results in Mathematics
Approximation by Multivariate
Max-Product Kantorovich Exponential
Sampling Operators
Sathish Kumar Angamuthu
Abstract. The approximation behavior of multivariate max-product Kan-
torovich exponential sampling operators has been analyzed. The point-
wise and uniform approximation theorem for these sampling series Iχ,(M)
w,j
is proved. The degree of approximation in-terms of logarithmic modulus
of smoothness is studied. For the class of log-H¨olderian functions, the
order of uniform norm convergence is established. The norm-convergence
theorems for the multivariate max-product Kantorovich exponential sam-
pling operators in Mellin–Lebesgue spaces is studied.
Mathematics Subject Classification. 41A35, 94A20, 41A25.
Keywords. Multivariate max-product operators, Kantorovich sampling
operators, degree of approximation, Mellin–Lebesgue spaces.
1. Introduction
One of the fundamental results in Fourier analysis is the celebrated Shannon
sampling theorem due to Wittaker, Kotelnikov and Shannon (see [29]). The
sampling theorem has played an important role in the areas of approximation
theory, harmonic analysis, signal and image processing etc., due to the fact
that it provides a tool to convert analog signals into a discrete sequence of
samples without loosing the actual information.Though WKS sampling theo-
rem is one of the most influential result in the theory of approximation but
certainly it requires strong assumptions on the functions to be approximated.
Butzer and Stens [19] initiated the study of generalized sampling series for
not necessarily band limited signals. They have replaced the sinc-function in
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
The Kantorovich exponential sampling series Iwχf at jump discontinuities of the bounded measurable signal f:R+→R has been analysed. A representation lemma for the series Iwχf is established and using this lemma certain approximation theorems for discontinuous signals are proved. The degree of approximation in terms of logarithmic modulus of smoothness for the series Iwχf is studied. Further a linear prediction of signals based on past sample values has been obtained. Some numerical simulations are performed to validate the approximation of discontinuous signals f by Iwχf.
Article
Full-text available
In this paper, we introduce a new family of operators by generalizing Kantorovich type of exponential sampling series by replacing integral means over exponentially spaced intervals with its more general analogue, Mellin Gauss Weierstrass singular integrals. Pointwise convergence of the family of operators is presented and a quantitative form of the convergence using a logarithmic modulus of continuity is given. Moreover, considering locally regular functions, an asymptotic formula in the sense of Voronovskaja is obtained. By introducing a new modulus of continuity for functions belonging to logarithmic weighted space of functions, a rate of convergence is obtained. Some examples of kernels satisfying the obtained results are presented.
Article
Full-text available
We analyze the behaviour of the exponential sampling series SwχfS_{w}^{\chi }f at jump discontinuity of the bounded signal f. We obtain a representation lemma that is used for analyzing the series SwχfS_{w}^{\chi }f and we establish approximation of jump discontinuity functions by the series Swχf.S_{w}^{\chi }f. The rate of approximation of the exponential sampling series SwχfS_{w}^{\chi }f is obtained in terms of logarithmic modulus of continuity of functions and the round-off and time-jitter errors are also studied. Finally we give some graphical representation of approximation of discontinuous functions by SwχfS_{w}^{\chi }f using suitable kernels.
Article
Full-text available
In the present article, we introduce and analyse the approximation properties of the new family of exponential sampling type neural network operators activated by the sigmoidal functions. We derive the pointwise and uniform convergence theorem and study the order of approximation for these family of operators. Further, we establish the quantitative estimate for the order of convergence in terms of modulus of continuity and also discuss the convergence of exponential sampling type quasi interpolation operators. At the end, we provide a few examples of the sigmoidal functions satisfying the presented theory.
Article
Full-text available
In this article, we analyse the behaviour of the new family of Kantorovich type exponential sampling series. We derive the point-wise approximation theorem and Voronovskaya type theorem for the series (Iwχ)w>0.(Iwχ)w>0.(I_{w}^{\chi })_{w>0}. Further, we establish a representation formula and an inverse result of approximation for these operators. Finally, we give some examples of kernel functions to which the theory can be applied along with the graphical representation.
Article
In this paper we study the convergence properties of certain semi-discrete exponential-type sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed.
Article
In this paper, we generalize the family of exponential sampling series for functions of n variables and study their pointwise and uniform convergence as well as the rate of convergence for the functions belonging to space of log-uniformly continuous functions. Furthermore, we state and prove the generalized Mellin-Taylor’s expansion of multivariate functions and using this expansion we establish pointwise asymptotic behaviour of the series by means of Voronovskaja type theorem.
Article
This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section of the function involved. Approximation behavior of newly con- structed operators is investigated at continuity points for log-uniformly continuous functions. The rate of convergence of the series is presented for the same functions by means of logarithmic modulus of continuity. A Voronovskaja type theorem is also presented by means of Mellin derivative.
Article
In the present article, we extend our study of Kantorovich type exponential sampling operators introduced in [4]. We derive the Voronovskaya type theorem and its quantitative estimates for these operators in terms of an appropriate K-functional. Further, we improve the order of approximation by using the convex type linear combinations of these operators. Finally, we provide few examples of kernels along with the graphical representations.
Chapter
In the present article, we extend the theory of exponential sampling type neural network operators to the max-product setting. The approximation properties of these operators activated by the sigmoidal functions have been studied by using the moment type approach. We establish the point-wise and uniform approximation theorem for these operators along with the quantitative estimate of the order of convergence using the modulus of continuity. Consequently, we discuss the convergence of exponential sampling type quasi-interpolation operators of the max-product kind. At the end, we provide a few examples of the sigmoidal functions satisfying the assumptions of the presented theory.