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Approximation properties of multivariate exponential sampling series
Abstract
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.
... They have replaced the sinc-function in where w > 0 and χ is kernel function satisfying the suitable assumptions. The approximation results of the above sampling series S χ w was analysed in several directions, see [4,6,7,12,13,30,31] etc. To approximate the Lebesgue integrable functions on R + , the Kantorovich version of S χ w was considered in [3]. ...
... For f ∈ C(I n ), the logarithmic modulus of smoothness is defined by (see [9] and [30]) ...
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) 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.
... Later on Kantorovich [12] versions were studied in previous studies [13][14][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]). However, this kind of applications is mainly meaningful in three-dimensional case. ...
... We used this expression in our simulations to estimate the quality of the results achieved by means of the Durrmeyer operators. In our reconstructions, the Durrmeyer operators have been equipped with the following discrete Mellin-spline (1 + log(i))(1 + log( )), i e −1 < i ≤ 1, e −1 < ≤ 1, (1 + log(i))(1 − log( )), i e −1 < i ≤ 1, 1 ≤ < e, (1 − log(i))(1 + log( )), i 1 ≤ i < e, e −1 < ≤ 1, (1 − log(i))(1 − log( )), i 1 ≤ i < e, 1 ≤ < e, 0, otherwise, (19) implemented in Matlab©, with i, ∈ N being the coordinates of the discrete grid considered for the calculation. In Algorithm 1, the pseudocode is shown. ...
In this paper, we study the convergence properties of certain semi‐discrete exponential‐type sampling series in a multidimensional frame. In particular, we obtain an asymptotic formula of Voronovskaya type, which gives a precise order of approximation in the space of continuous functions, and we give some particular example illustrating the theory. Applications to the study of the seismic waves are illustrated.
... where f : R + → R such that the above series is absolutely convergent. The convergence properties of S ϕ w were analysed by many researchers, see [13,15,25,26,40,41] and the references therein. The above sampling series S ϕ w is not suitable to approximate integrable functions on R + . ...
The convergence in variation for the exponential sampling operators and Kantorovich exponential sampling operators based on averaged-type kernel has been analyzed. A characterization of the space of absolutely continuous functions in terms of the convergence in variation by means of these exponential sampling operators is studied.
... Similarly, generalized Durrmeyer sampling operators (see, [11,21]) provide a further generalization by integrating polynomial terms. Additional developments, such as exponential sampling series and modifications introduced by Bardaro et al. [13] (for further see, [2,3,28,31]), have broadened the applicability of these methods. These advances have enabled the study of phenomena like light scattering and diffraction. ...
The present paper deals with construction of newly family of Neural Network operators, that is,Steklov Neural Network operators. By using Steklov type integral, we introduce a new version of Neural Network operators and we obtain some convergence theorems for the family, such as, pointwise and uniform convergence,rate of convergence via moduli of smoothness of order r.
... In multivariate settings, H. Karsli, [31] and U. Kadak, [30] give the Urysohn type and fractional sampling Kantarovich operators, respectively. Meanwhile, the approximation properties using multivariate exponential sampling operators has been elaborated in [33]. Again, Costarelli et al. derived theorems and corollaries regarding Durremeyer-sampling type of operators [26] in Orlicz space and recently, a modified version of sampling series [37] has been derived by M. Turgay and T. Acar. ...
This paper investigates the convergence of a family of multivariate exponential Durrmeyer-type sampling type operators within the framework of the logarithmic modulus of continuity. Additionally, the study establishes Voronovskaja type formula in Mellin setting for the rate of convergence results, accompanied by the quantitative estimates in terms of moments corresponding to the defined kernel functions.
... Numerous results have been published on the exponential sampling series and Kantorovich forms; see, for example, [32][33][34][35][36][37][38][39][40][41][42]. For other publishes on approximation theory and sampling type series, we refer the readers to [43,44]. ...
In the present paper, we analyze the behavior of the exponential‐type generalized sampling Kantorovich operators Kωφ,G when discontinuous signals are considered. We present a proposition for the series Kωφ,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.
... Recently, numerous results have been published on the exponential sampling series and its different forms (see, e.g., [5][6][7][8][9][14][15][16][17][18]25,26,51]). In the last century, there has been increasing interest in fractional-type calculus and its applications. ...
The present paper deals with construction of a new family of exponential sampling Kantorovich operators based on a suitable 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 presented and the results are examined by illustrative numerical tables and graphical representations.
... The simultaneous approximation properties and the direct and inverse results of (2) were studied by Heilmann in [30] and [32] respectively. The Durrmeyer variants of various linear positive operators have been thoroughly studied by researchers in [2], [9], [10], [11], [17], [35], [37], [39], [43], [44] and [48]. Further, direct and convergence results on weighted simultaneous approximation have been obtained by Heilmann and Müller in [31]. ...
... Then, they investigated essential convergence results of them. The family of operators (2) has been studied by considering its different forms: Kantorovich forms in [7], Durrmeyer forms in [14], bivariate forms in [9], multivariate forms in [4,30]. Another recent study on exponential sampling series is due to Aral et al. [8] in which authors constructed 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. ...
... Moreover, the approximation behaviour of the series (1.1) in the setting of Mellin-Lebesgue spaces was analyzed in [16]. The approximation properties of bivariate and multivariate signals by S χ w were discussed in [6] and [32] respectively. The approximation results of neural network exponential sampling series were considered in [4]. ...
In the present article, an inverse approximation result and saturation order for the Kantorovich exponential sampling series are established. First we obtain a relation between the generalized exponential sampling series and for the space of all uniformly continuous and bounded functions on Next, a Voronovskaya type theorem for the sampling series is proved. The saturation order for the series is obtained using the Voronovskaya type theorem. Further, an inverse result for is established for the class of log-H\"{o}lderian functions. Moreover, some examples of kernels satisfying the conditions, which are assumed in the hypotheses of the theorems, are discussed.
... A bivariate extension of the generalized exponential sampling operators (1.1) in order to give applications to the study of the seismic waves was given in [6]. A multivariate forms and multivariate Kantorovich forms of (1.1) have been introduced and studied in [27] and [2], respectively. Further, in the recent paper [15] Bardaro et al. introduced Durrmeyer modification of the operators (1.1). ...
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.
In this article, we study the convergence behaviour of the classical generalized Max Product exponential sampling series in the weighted space of log-uniformly continuous and bounded functions. We derive basic convergence results for both the series and study the asymptotic convergence behaviour. Some quantitative approximation results have been obtained utilizing the notion of weighted logarithmic modulus of continuity.
In the present paper, we introduce a new family of sampling operators, so-called “modified sampling operators”, by taking a function ρ that satisfies the suitable conditions, and we study pointwise and uniform convergence of the family of newly introduced operators. We give the rate of convergence of the family of operators via classical modulus of continuity. We also obtain an asymptotic formula in the sense of Voronovskaja. Moreover, we investigate the approximation properties of modified sampling operators in weighted spaces of continuous functions characterized by ρ function. Finally, we present examples of some kernels that satisfy the appropriate assumptions. At the end, we present some graphical and numerical representations by comparing the modified sampling operators and the classical sampling operators.
In this paper, we investigate the approximation properties of exponential sampling series within logarithmically weighted spaces of continuous functions. Initially, we demonstrate the pointwise and uniform convergence of exponential sampling series in weighted spaces and present the rates of convergence via a suitable modulus of continuity in logarithmic weighted spaces. Subsequently, we establish a quantitative representation of the pointwise asymptotic behavior of these series using Mellin–Taylor’s expansion. Finally, it is given some examples of kernels and numerical evaluations.
In this paper, we introduce a family generalized Kantorovich-type exponential sampling operators of bivariate functions by using the bivariate Mellin-Gauss-Weierstrass operator. Approximation behaviour of the series is established at continuity points of log-uniformly continuous functions. A rate of convergence of the family of operators is presented by means of logarithmic modulus of continuity and a Voronovskaja-type theorem is proved in order to determine rate of pointwise convergence. Convergence of the family of operators is also investigated for functions belonging to weighted space. Furthermore, some examples of the kernels which support our results are given.
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.
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