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Approximation by bivariate generalized sampling series in weighted spaces of functions
... For other publications in the literature on this subject, see also [3,4,9,17,18,29-34, 36, 45]. Furthermore, for the approximation properties of generalized sampling operators and their different forms in weighted spaces of continuous functions (see, [1,2,8,10,43]). ...
In this paper, we introduce Mellin-Steklov exponential samplingoperators of order , by considering appropriate Mellin-Steklov integrals. We investigate the approximation properties of these operators in continuousbounded spaces and spaces on By using the suitablemodulus of smoothness, it is given high order of approximation. Further, we present a quantitative Voronovskaja type theorem and we study the convergence results of newly constructed operators in logarithmic weighted spaces offunctions. Finally, the paper provides some examples of kernels that support the our results.
... where is an locally integrable function defined on R such that the series is absolutely convergent for every x ∈ R; see [5]. For the rest of the works in classical sampling and its Kantorovich forms, for example, see [6][7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein. Freud [20] examined the widely recognized Hermite-Fejer interpolation process, H n , applied at the zeros of general orthogonal polynomials and provided criteria to guarantee that lim n→∞ H n (x) = (x). ...
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.
... For the rest of the sampling theory, see [45,59]. In the field of classical sampling theory, numerous results have been published in the papers [1][2][3][4][10][11][12]20,[40][41][42]57]. ...
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.
... In order to determine the rate of convergence, a new modulus of continuity, called "weighted logarithmic modulus of continuity" was introduced. Here we mention that weighted approximation of sampling type operators are very recent and active research area, for most recent paper on weighted approximation of classical sampling operators, we refer the readers to [1,2,5,6]. However, rate of pointwise convergence and an upper estimate for pointwise convergence were not presented for the operators (3). ...
In this article, we introduce new generalization of Szász–Mirakjan operators. First, we give the recurrence relationship for the moments of these operators and present the central moments up to the fourth degree. Then, we give the local approximation properties of these operators through Peetre’s K-function. Furthermore, we investigate rate of convergence by using modulus of continuity and Lipschitz type maximal functions. Then, we investigate approximation properties of these operators on weighted space. Also, we prove voronoskaja type theorem. Additionally, the bivariate type of these operators is introduced, and the approximate behaviors of these operators are examined. Finally, we give some numerical illustrative examples.
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.
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