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Approximation by sampling Kantorovich series in weighted spaces of functions
... Remark 5.5. The convergence results established in the present paper extend for the nonlinear sampling Kantorovich operators (and also for the linear ones, see Remark 3.1) the convergence results proved in [24]. More precisely, in [24] the convergence has been proved for the usual weighted sup-norm, while here we can deduce the convergence with respect to the weighted L p -norm, 1 ≤ p < +∞. ...
... The convergence results established in the present paper extend for the nonlinear sampling Kantorovich operators (and also for the linear ones, see Remark 3.1) the convergence results proved in [24]. More precisely, in [24] the convergence has been proved for the usual weighted sup-norm, while here we can deduce the convergence with respect to the weighted L p -norm, 1 ≤ p < +∞. ...
In the present paper, convergence in modular spaces is investigated for a class of nonlinear discrete operators, namely the nonlinear multivariate sampling Kantorovich operators. The convergence results in the Musielak-Orlicz spaces, in the weighted Orlicz spaces, and in the Orlicz spaces follow as particular cases. Even more, spaces of functions equipped by modulars without an integral representation are presented and discussed.
... 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. ...
... 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). ...
... 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.
... The proofs of these theorems are omitted here. For further details, readers are referred to the original sources (see [1,2]). ...
In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions.
... numerical analysis, and optimization problems (see [3][4][5][6][7][8][9] and [10]). The classical Bernstein operators are a widely-used tool for approximating functions in the continuous function space on the interval [0, 1] (see [2,[11][12][13][14][15] and [16]). ...
In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function ψα and investigate their approximation properties. By choosing an appropriate function ψα, the order of approximation of our operators to a function f f is at least as good as the classical Bernstein–Kantorovich operators on the interval [0,1]. We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided.
... Acar and Draganov [31] established a strong converse inequality for the rate of convergence of generalized sampling operators with the aid of moduli of smoothness. Acar et al. [32] derived the quantitative estimates for the rate of approximation of sampling Kantorovich operators in terms of weighted modulus of continuity for continuous functions belonging to weighted spaces and also established Voronovskaja type theorems in quantitative form. Recently, Acar and Draganov [33] presented a strong converse inequality for the rate of the simultaneous approximation by generalized sampling operators in the L p -norm, 1 ≤ p ≤ ∞. ...
In the present article, we introduce a Kantorovich variant of the neural network interpolation operators activated by smooth ramp functions proposed by Qian and Yu (2022). We discuss the convergence of these operators in the spaces, C([c,d]) and Lp([c,d]),1≤p<∞, and establish some direct approximation theorems. Further, we derive the converse results by means of Berens–Lorentz lemma and Peetre's K‐functional. We present a multivariate version of the aforementioned Kantorovich neural network interpolation operators and investigate the direct and converse results in the continuous and Lp,1≤p<∞, spaces.
... 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.
... In recent years, Kantorovich-type generalizations have been a popular area of research both in the sense of defining new operators as well as in respect of exponential sampling series. For further investigation, one can see, [2][3][4][5]7,8,16,[18][19][20][21]30] and the reference cited therein. ...
In the present paper, we introduce a new sequence of Bernstein-Kantorovich type operators, which fix constant and preserve Korovkin’s other test functions in a limiting sense. We extend the natural Korovkin and Voronovskaja type results into a sequence of probability measure spaces. Then, we establish the convergence properties of these operators using the Ditzian-Totik modulus of smoothness for Lipschitz-type space and functions with derivatives of bounded variations.
... 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.
... Subsequently, some preservation properties of the Baskakov-Kantorovich operator were studiedly Zhang in [54]. For recent developments in this direction, we refer to [1], [4], [5], [6], [13], [22], [24] and [36]. ...
... One can think to extend such results to more general families of operators and the more general context of weighted spaces presented in the papers [15,16]. A significant direction of research in the Approximation Theory by positive linear operators is concerned with the complete asymptotic expansion of such operators (see [17][18][19][20][21]). ...
In a series of papers, most of them authored or co-authored by H. Gonska, several authors investigated problems concerning the composition and decomposition of positive linear operators defined on spaces of functions. For example, given two operators with known properties, A and B, we can find the properties of the composed operator A∘B, such as the eigenstructure, the inverse, the Voronovskaja formula, and the second-order central moments. One motivation for studying composed operators is the possibility to obtain better rates of approximation and better Voronovskaja formulas. Our paper will address such problems involving compositions of some classical positive linear operators. We present general results as well as numerical experiments.
... Estimates of the rate of approximation of general sampling operators (1.1) in spaces of continuous functions associated with the weight ρ 2,2 have been recently obtained in [1]. Similar results for integral modifications of the general sampling operator were established in [2,3]. Also, such results were proved for an integral form of general exponential sampling operators in function spaces equipped with a logarithmic weight in [4,Section 5]. ...
We construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a K-functional. The case of weighted approximation is also considered. The weights are positive and power-type with non-positive exponents at infinity. This sampling operator preserves every algebraic polynomial.
... Similar considerations for generalized sampling Kantorovich series and generalized exponential sampling series were studied in refs. [17] and [18], respectively. ...
The present article deals with local and global approximation behaviors of sampling Durrmeyer operators for functions belonging to weighted spaces of continuous functions. After giving some fundamental notations of sampling type approximation methods and presenting well definiteness of the operators on weighted spaces of functions, we examine pointwise and uniform convergence of the family of operators and determine the rate of convergence via weighted modulus of continuity. A quantitative Voronovskaja theorem is also proved in order to obtain rate of pointwise convergence and upper estimate for this convergence. The last section is devoted to some numerical evaluations of sampling Durrmeyer operators with suitable kernels.
The motive of this paper is to introduce the generalization of Lupaş-Kantorovich operators connected with Pólya distribution and establish the rate of convergence in terms of modulus of continuity. Furthermore, a Voronovskaja type asymptotic formula for these operators is studied. In the end, few numerical examples with graphical representation are added to depict the effect of convergence of the operators.
In this paper, we discuss the higher-order -Bernstein–Kantorovich operators, which are associated with the Bernstein polynomials. We analyze various aspects, such as error estimations and Voronovskaja-type asymptotic formula. Finally, we provide a comparative study by visualizing the results graphically and interpreting the upper limit of the error numerically.
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 work discusses two periods of transformation of the Proto-Kartvelian population: the first (L–XXV centuries BC) when the entire population spoke one Proto-Kartvelian language and lived in a relatively large area; the second period – (XXV–X centuries BC), when the population divided into three parts: Proto-Svan; speaking the Colchian-Georgian language and the third part was
scattered on the European continent.
The second period is described by two different mathematical models: a part of the Proto-Kartvelian speaking population went to Europe and slowly began the process of their assimilation on the European continent. The unknown function that determines the number of Proto-Kartvelian speaking people in Europe at the time is described by a Pearl - Verhulst-type mathematical model with variable coefficients that also take the assimilation process into account. The analytical
solution of the Cauchy problem is found in quadratures.
The population that remained primarily in former Asia and the Caucasus region was gradually divided into two groups: those who spoke the Proto-Svan and those who spoke Colchian-Georgian languages. To describe their interference and development, a mathematical model is used, which is described by a nonlinear dynamic system with nonlinear terms of self-limitation and takes into
account the unnatural reduction of the Colchian-Georgian population as a result of hostilities with neighboring peoples. For a dynamic system without nonlinear terms of self-constraint, in the case of certain relationships between variable coefficients, the first integral was found, by means of which the Bernoulli equation with variable coefficients was obtained for one of the unknown functions. In the case of constant coefficients of the dynamic system, for certain dependences
between the coefficients, the dynamic system follows the system of Lotka–Volterra equations, with corresponding periodic solutions.
For the general mathematical model (nonlinear terms of self-limitation and unnatural reduction of the Colchian-Georgian population due to hostilities with neighboring peoples) in two cases of certain interdependencies between constant coefficients, it is shown that the divergence of an unknown vector-function in the physically meaningful first quarter of the phase plane changes the sign when passing through some half-direct one. Taking into account the principle (theorem) of Bendixson, theorems have been proved, on the variability of the divergence of the vector field and the existence of closed trajectories in some singly connected domain of the point located on this half-direct (starting point of the trajectory).
Thus, for the general dynamic system, with some dependencies between constant coefficients, it is shown that there is no assimilation of the Proto-Svan population by the Colchian-Georgian population and these two populations (Colchian-Georgian and Proto-Svan) coexist peacefully in virtually the same region due to the transformation of the Proto-Kartvelian population.
The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is proved for functions belonging to weighted spaces. A rate of convergence by means of weighted moduli of continuity is presented and a quantitative Voronovskaja type theorem is obtained.
In the present paper, we study the saturation order in the space for the sampling Kantorovich series based upon bandlimited kernels. The above study is based on the so-called Fourier transform method, introduced in 1960 by P.L. Butzer. As a first result, the saturation order is derived in a Bernstein class; here, it is crucial to derive the Fourier transform of the above sampling-type series, which can be expressed in a suitable closed form. Subsequently, the saturation is reached in the whole space . At the end of the paper, several examples of bandlimited kernels, such as the Fej\'er's and Bochner-Riesz's kernel, have been recalled and the saturation order of the corresponding sampling Kantorovich series has been stated.
In this paper, we study the convergence in variation for the generalized sampling operators based upon averaged-type kernels and we obtain a characterization of absolutely continuous functions. This result is proved exploiting a relation between the first derivative of the above operator acting on f and the sampling Kantorovich series of f'. By such approach, also a variation detracting-type property is established. Finally, examples of averaged kernels are provided, such as the central B-splines of order n (duration limited functions) or other families of kernels generated by the Fejer and the Bochner-Riesz kernels (bandlimited functions).
By using the so-called max-product method, in this paper we associate to the truncated Whittaker linear operator and to the non-truncated and truncated linear sampling operators based on the Fejér-type kernel, nonlinear sampling operators for which Jackson-type approximation orders in terms of the first order modulus of continuity are obtained, thus essentially improving the orders of approximation given by their linear correspondents. Also, we prove that the max-product truncated Whittaker operator and the max-product truncated operator based on the Fejér-type kernel, partially preserve the monotonicity.
In this paper, the problem of the order of approximation for the multivariate
sampling Kantorovich operators is studied. The cases of the uniform
approximation for uniformly continuous and bounded functions/signals belonging
to Lipschitz classes and the case of the modular approximation for functions in
Orlicz spaces are considered. In the latter context, Lipschitz classes of
Zygmund-type which take into account of the modular functional involved are
introduced. Applications to Lp(R^n), interpolation and exponential spaces can
be deduced from the general theory formulated in the setting of Orlicz spaces.
The special cases of multivariate sampling Kantorovich operators based on
kernels of the product type and constructed by means of Fejer's and B-spline
kernels have been studied in details.
The aim of this note is that by using the so-called max-product method to associate to some generalized sampling approximation linear operators and to the Whittaker cardinal series, nonlinear sampling operators for which Jackson-type approximation orders in terms of the moduli of smoothness are obtained. In the case of max-product Whittaker operator, for positive valued functions, essentially a better order of approximation is obtained.
In 1903 Fredholm published his famous paper on integral equations. Since then linear integral operators have become an important tool in many areas, including the theory of Fourier series and Fourier integrals, approximation theory and summability theory, and the theory of integral and differential equations. As regards the latter, applications were soon extended beyond linear operators. In approximation theory, however, applications were limited to linear operators mainly by the fact that the notion of singularity of an integral operator was closely connected with its linearity. This book represents the first attempt at a comprehensive treatment of approximation theory by means of nonlinear integral operators in function spaces. In particular, the fundamental notions of approximate identity for kernels of nonlinear operators and a general concept of modulus of continuity are developed in order to obtain consistent approximation results. Applications to nonlinear summability, nonlinear integral equations and nonlinear sampling theory are given. In particular, the study of nonlinear sampling operators is important since the results permit the reconstruction of several classes of signals. In a wider context, the material of this book represents a starting point for new areas of research in nonlinear analysis. For this reason the text is written in a style accessible not only to researchers but to advanced students as well.
Contents
1. Kernel functionals and modular spaces
2. Absolutely continuous modulars and moduli of continuity
3. Approximation by convolution type operators
4. Urysohn integral operators with homogeneous kernel functions. Applications to nonlinear Mellin-type convolution operators
5. Summability methods by convolution-type operators
6. Nonlinear integral operators in the space BV[subscript [actual symbol not reproducible]]
7. Application to nonlinear integral equations
8. Uniform approximation by sampling type operators. Applications in signal analysis
9. Modular approximation by sampling type operators.
This paper deals with the Kantorovich version of generalized sampling series, the first one to be primarily concerned with this version. It is devoted to the study of these series on Orlicz spaces, L φ (ℝ), in the instance of irregularly spaced samples. A modular convergence theorem for functions f∈L φ (ℝ) is deduced. The convergence in L p (ℝ)-space, LlogL-space, and exponential spaces follow as particular results. Applications are given to several sampling series with special kernels, especially in the instance of discontinuous signals. Graphical representations for the various examples are included.
There are several reasons why the classical sampling theorem is rather impractical for real life signal processing. First, the sinc-kernel is not very suitable for fast and efficient computation; it decays much too slowly. Second, in practice only a finite number N of sampled values are available, so that the representation of a signal f by the finite sum would entail a truncation error which decreases rather slowly for N¿ ¿, due to the first drawback. Third, band-limitation is a definite restriction, due to the nonconformity of band and time-limited signals. Further, the samples needed extend from the entire past to the full future, relative to some time t = t0. This paper presents an approach to overcome these difficulties. The sinc -function is replaced by certain simple linear combinations of shifted B-splines, only a finite number of samples from the past need be available. This deterministic approach can be used to process arbitrary, not necessarily bandlimited nor differentiable signals, and even not necessarily continuous signals. Best possible error estimates in terms of an Lp-average modulus of smoothness are presented. Several typical examples exhibiting the various problems involved are worked out in detail.
In this paper the behaviour of the first derivative of the so-called sampling Kantorovich operators has been studied, when both differentiable and not differentiable signals are taken into account. In particular, we proved that the family of the first derivatives of the above operators converges pointwise at the point of differentiability of f, and uniformly to the first derivative of a given C1 signal. Moreover, the convergence has been analysed also in presence of points in which a signal is not differentiable. In the latter case, under suitable assumptions, it can be proved the pointwise convergence of the first derivative of the above operators to a certain combination of the left and right first derivatives (whenever they exist and are both finite) of the signals under consideration. Finally, also the extension of the above results to the case of signals with high order derivatives has been discussed. At the end of the paper, examples of kernel functions and graphical representations are provided. Further, also numerical examples in the cases of functions for which the left and right derivatives are ±∞ or they do not exist in R˜, have been given.
In this paper we continue the study of certain Bernstein-Chlodovsky operators Bn⁎ preserving the exponential function e−2x (x≥0), recently introduced in [4]. In particular, we prove some Voronovskaya type theorems and we deduce some properties of the Bn⁎'s, such as saturation results. We also compare this new class of operators with the classical Bernstein-Chlodovsky ones, proving that the operators Bn⁎ provide better approximation results for certain functions.
In the present paper, a new family of sampling type operators is introduced and studied. By the composition of the well-known generalized sampling operators of P.L. Butzer with the usual differential and anti-differential operators of order m, we obtain the so-called m-th order Kantorovich type sampling series. This family of approximation operators are very general and include, as special cases, the well-known sampling Kantorovich and the finite-differences operators. Here, we discuss about pointwise and uniform convergence of the m-th order Kantorovich type sampling series; further, quantitative estimates for the order of approximation have been established together with asymptotic formulas and Voronovskaja type theorems. In the latter results, a crucial role is played by certain algebraic moments of the involved kernels, that can be computed by resorting to the their Fourier transform and to the well-known Poisson’s summation formula. By means of the above results we become able to solve the problems of the simultaneous approximation of a function and its derivatives, both from a qualitative and a quantitative point of view, and of the linear prediction by samples from the past.
In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function e^-2x (x>0). We study its approximation properties in several function spaces, and we evaluate the rate of convergence by means of suitable moduli of continuity. Throughout some estimates of the rate of convergence, we prove better error estimation than the original operators on certain intervals.
In the present paper, we study the saturation order for the sampling Kantorovich series in the space of uniformly continuous and bounded functions. In order to achieve the above result, we first need to establish a relation between the sampling Kantorovich operators and the classical generalized sampling series of P.L. Butzer. Further, for the latter operators, we also need to prove a Voronovskaja-type formula. Moreover, inverse theorems of approximation are also obtained, in order to give a characterization for the rate of convergence of the above operators.
Finally, several examples of kernel functions for which the above theory can be applied have been presented and discussed in details.
In this paper we obtain the saturation order, a strong localization result and local saturation results in the approximation of continuous positive functions by the truncated max-product sampling operators based on sinc (Wittaker-type) and Fejér-type kernels. The localization-type results obtained present important potential applications in signal theory.
In this paper, the behavior of the sampling Kantorovich operators has been studied, when discontinuous functions (signals) are considered in the above sampling series. Moreover, the rate of approximation for the family of the above operators is estimated, when uniformly continuous and bounded signals are considered. Finally, several examples of (duration-limited) kernels which satisfy the assumptions of the present theory have been provided, and also the problem of the linear prediction by sampling values from the past is analyzed.
If a signal f is band-limited to [—πW, πW] for some W > 0, then f can be completely reconstructed for all values of t ∈ R from its sampled values f (k/W), k ∈ Z, taken just at the nodes k/W, equally spaced apart on the whole R, in terms of \begin{array}{*{20}{c}} {f(t) = \sum\limits_{{k = - \infty }}^{\infty } {f\left( {\frac{k}{W}} \right)\frac{{\sin \pi (Wt - k)}}{{\pi (Wt - k)}}} } & {(t \in R).} \\ \end{array}
(5.1.1)
KeywordsBounded Linear OperatorLinear PredictionPolynomial SplinePast SampleNyquist RateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor’s formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor’s formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Grüss type approximation theorem and the Grüss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.
This paper deals with approximations of bounded, uniformly continuous on the whole real axis, functions by the interpolating Shannon sampling operators. These interpolating operators are defined by the kernels obtained by dilation from known ones, in particular, such as Rogosinski and Blackman-Harris kernels. The focus of our study is on estimates for operator norms and approximation orders via moduli of continuity.
Here we give some pointwise convergence theorems and asymptotic formulae of Voronovskaja type for a general class of Kantorovich discrete operators. Applications to the Kantorovich version of some discrete operators are given.
In this paper we consider some generalized Shannon sampling operators, which are defined by band-limited kernels. In particular, we use dilated an averaged versions of some previously known kernels. We give also some examples of using sampling operators in imaging applications.
In this paper an asymptotic formula of Voronovskaja type for a multivariate extension of the Kantorovich generalized sampling series is given. Moreover a quantitative version in terms of some moduli of smoothness is established. Finally some particular examples of kernels are discussed, as the Bochner-Riesz kernel and the multivariate splines. © 2011 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.
This note is concerned with the truncation errors for generalized sampling series, which are defined by the Hann and Blackman
window functions.
On approximation properties of sampling operators by dilated kernels
- A Kivinukk
- G Tamberg
Kivinukk A, Tamberg G. On approximation properties of sampling operators by dilated kernels. In: 8th International
Conference on Sampling Theory and Applications (SAMPTA'09); Marseille, France; 2009.
On approximation properties of generalized Kantorovich-type sampling operators
- O Orlova
- G Tamberg
Orlova O, Tamberg G. On approximation properties of generalized Kantorovich-type sampling operators. Journal
of Approximation Theory 2016; 201: 73-86. doi: 10.1016/j.jat.2015.10.001