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Approximation Properties of Exponential Sampling Series in Logarithmic Weighted Spaces
Abstract
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.
... The generalized sampling operator and its various forms, as well as the exponential sampling operator and its variants, are not only theoretically significant but also play a role in modeling seismic waves (see, e.g., [10,11]). Further, very recently, in papers [3,4,6], the authors extended (log) continuous spaces of functions by considering the weight function 1 + log 2 (·) on R + and investigated the approximation properties of the series given by (1.5) and its different forms in logarithmic weighted spaces of functions. ...
In this study, we begin by deriving a two-term strong converse inequality that characterizes the rate of approximation for generalized exponential sampling operators, utilizing the logarithmic moduli of smoothness. Subsequently, we combine the direct and converse estimates to establish the saturation property and identify the class of these approximation operators. Finally, we provide the results for the generalized exponential sampling series using specific kernels that meet the necessary assumptions. It should be noted that in the present study we follow a direct approach within the framework of Mellin analysis in the processes obtained.
... For other publications on the exponential sampling series and its different forms (see, [12][13][14][15][40][41][42]). Moreover, for the approximation properties of generalized exponential sampling series and its different forms in logarithmic weighted spaces of continuous functions (see also [5][6][7]). ...
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.
... 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.
... 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.
... In recent years, numerous studies have been published on sampling type operators. We can refer the readers to [19,21,28,39] for generalized sampling operators, [9,11,26] for sampling Kantorovich operators, [14,24] for sampling Durrmeyer operators, [5,6,12,37] for exponential sampling type operators and [ 3,4,8,10,[34][35][36]38] for both polynomial and logarithmic weighted approximation by sampling type operators. Our aim in this paper is to construct a new form of generalized sampling operators given in (1.1) by considering a ρ function, which satisfies some suitable conditions. ...
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.
... 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.
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.
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.
This paper deals with the study of the convergence of the family of multivariate Durrmeyer-sampling type operators in the general setting of Orlicz spaces. The above result implies also the convergence in remarkable subcases, such as in Lebesgue, Zygmund and exponential spaces. Convergence results have been established also in case of continuous functions, where pointwise and uniform convergence theorems, including some quantitative estimates, have been achieved. Finally, several examples with graphical representations are given.
In real world applications, signals can be suitably reconstructed by nonlinear procedures; this justifies the study of nonlinear approximation operators. In this paper, we prove some quantitative estimates for the nonlinear sampling Kantorovich operators in the multivariate setting using the modulus of smoothness of L p (n). The above results have been then extended to the general case of Orlicz spaces L ϕ (n), so obtaining quantitative estimates in several instances of well-known and useful spaces.
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.
In this paper, we establish a quantitative estimate for Durrmeyer-sampling type operators in the general framework of Orlicz spaces, using a suitable modulus of smoothness defined by the involved modular functional. As a consequence of the above result, we can deduce quantitative estimates in several instances of Orlicz spaces, such as L p -spaces, Zygmund spaces and the exponential spaces. By using a direct approach, we also provide a further estimate in the particular case of L p -spaces, with 1 ≤ p < + ∞ , that turns out to be sharper than the previous general one. Moreover, we deduce the qualitative order of convergence, when functions belonging to suitable Lipschitz classes are considered.
We continue the work started in a previous article and introduce a general setting in which we define nets of nonlinear operators whose domains are some set of functions defined in a locally compact topological group. We analyze the behavior of such nets and detect the fairest assumption, which are needed for the nets to converge with respect to the uniform convergence and in the setting of Orlicz spaces. As a consequence, we give results of convergence in this frame, study some important special cases, and provide graphical representations.
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 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 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 paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of smoothness in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of Lp-spaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the general case.
In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.
Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on , and in this context we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Then we obtain a modular convergence theorem in the general setting of Orlicz spaces . From the latter result, the convergence in -space, , and the exponential spaces follow as particular cases. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.
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. 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.
In this paper we introduce the generalized exponential sampling series of bivariate functions and establish some pointwise and uniform convergence results, also in a quantitative form. Moreover, we study the pointwise asymptotic behaviour of the series. One of the basic tools is the Mellin--Taylor formula for bivariate functions, here introduced. A practical application to seismic waves is also outlined.
Here we give asymptotic formulae of Voronovskaja type for linear combinations of exponential sampling series. Moreover we give a quantitative version in terms of some moduli of smoothness. Some examples are given.
In this paper we study norm-convergence to a function f of its generalized exponential sampling series in weighted Lebesgue spaces. Key roles are taken by a result on the norm-density of the test functions and the notion of bounded coarse variation. Some examples are described.
The purpose of this paper is to construct a bivariate generalization of new family of Kantorovich type sampling operators First, we give the pointwise convergence theorem and a Voronovskaja type theorem for these Kantorovich generalized sampling series. Further, we obtain the degree of approximation by means of modulus of continuity and quantitative version of Voronovskaja type theorem for the family Finally, we give some examples of kernels such as box spline kernels and Bochner–Riesz kernel to which the theory can be applied.
In the present paper, an inverse result of approximation, i.e., a saturation theorem for the sampling Kantorovich operators is derived, in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, here we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by P.L. Butzer. At the end, some other applications of such representation formula are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.
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.
In this paper, we study the problem of the rate of approximation for the family of sampling Kantorovich operators in the uniform norm, for uniformly continuous and bounded functions belonging to Lipschitz classes (Zygmund- type classes), and for functions in Orlicz spaces. The general setting of Orlicz spaces allows us to directly deduce the results concerning the order of approximation in Lp-spaces, 1 ≤ p < ∞, very useful in applications to Signal Processing, in Zygmund spaces and in exponential spaces. Particular cases of the sampling Kantorovich series based on Fejér's kernel and B-spline kernels are studied in detai.
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.
In this paper we study a linear version of the sampling Kantorovich type operators in a multivariate setting and we show applications to Image Processing. By means of the above operators, we are able to reconstruct continuous and uniformly continuous signals/images (functions). Moreover, we study the modular convergence of these operators in the setting of Orlicz spaces L W (R n) that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in L p
R n -spaces, L log L
R n -spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and Image Processing applications are included.
A powerful new method of inversion of the Laplace transform, based on its recently discovered eigenvalues and eigenfunctions, is applied to the problem of inverting light scattering data from a polydisperse molecular suspension.
With much material not previously found in book form, this book fills a gap by discussing the equivalence of signal functions with their sets of values taken at discreet points comprehensively and on a firm mathematical ground. The wide variety of topics begins with an introduction to the main ideas and background material on Fourier analysis and Hilbert spaces and their bases. Other chapters discuss sampling of Bernstein and Paley-Wiener spaces; Kramer's Lemma and its application to eigenvalue problems; contour integral methods including a proof of the equivalence of the sampling theory; the Poisson summation formula and Cauchy's integral formula; optimal regular, irregular, multi-channel, multi-band and multi-dimensional sampling; and Campbell's generalized sampling theorem. Mathematicians, physicists, and communications engineers will welcome the scope of information found here.
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.
We establish a two-term strong converse inequality for the rate of approximation of generalized sampling operators by means of the classical moduli of smoothness. It matches an already known direct estimate. We combine the direct and the converse estimates to derive the saturation property and class of this approximation operator. We demonstrate the general results for the sampling operators generated by the central B-splines, linear combinations of translates of B-splines, and the Bochner–Riesz kernel.
A read-only version of the paper is available at the following link, provided by Springer Nature SharedIt,
https://rdcu.be/cMw61
In the present article, we derive some approximation results concerning the order of convergence for a family of Durrmeyer type exponential sampling operators. Further, we improve the rate of approximation by constructing linear combinations of these operators. At the end, we provide a few examples of the kernel functions to which the presented theory can be applied along with the graphical representation.
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.
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.
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.
In this paper, we derive an inverse result for bivariate Kantorovich type sampling series for (the space of all continuous functions with upto second order partial derivatives are continuous and bounded on Further, we introduce the generalized Boolean sum (GBS) operators of bivariate Kantorovich type sampling series. We also study the rate of approximation for the GBS operators in terms of mixed modulus of smoothness and mixed K-functionals for the space of Bögel-continuous functions. Finally, we give some examples for the kernel to which the theory can be applied along with the graphical representations.
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.
Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin band-limited functions using both classical and generalized exponential sampling series.
In this paper we consider a new definition of generalized sampling type series using an approach introduced by Durrmeyer for the Bernstein polynomials. We establish an asymptotic formula for functions f with a polynomial growth and as a consequence we obtain a Voronovskaja type formula. Then we consider suitable linear combinations that provide a better order of approximation. Finally, some examples are given, in particular certain central B-splines are discussed.
We give for generalized Durrmeyer type series and their linear combinations quantitative Voronosvskaja formulae in terms of the classical Peetre K-functional. Finally we apply the general theory to various kernels
Illuminate an object with laser light and look at it through a diffraction grating. You will see a set of mutually displaced copies of the object. If the lateral extent of the object is small enough, the various copies do not overlap. You can easily devise a method for selecting a single copy of the object. For example, you can replace your own optical system, i.e., your eye, by a converging lens and let the multiple images that were impressing your retina be produced on a screen. Then, a hole on the screen will suffice to isolate a single image. You can even dispense with the laser light and repeat the observation in a more domestic environment by looking at a distant street lamp through a piece of fine fabric. In this case, all of the object copies except the central one will appear iridescent, but the basic phenomena will be the same.
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.
The aim of this paper is to present the counterpart of the theory of Fourier series in the Mellin setting in a systematic form, independently of the Fourier theory, under natural and minimal assumption upon the functions in question. Such Mellin (Fourier) series will be defined for functions f:ℝ + →ℂ for which f(x)=e 2πc f(e 2π x) for c∈ℝ and all x∈ℝ + , to be called c-recurrent functions, the counterpart of the periodic functions if c=0. The coefficients of the Mellin series or the finite Mellin transform will be connected with the classical Mellin transform on ℝ + by our Mellin-Poisson summation formula; it is the analogue of the famous Poisson summation formula connecting the Fourier transform on ℝ with the Fourier-coefficients or the finite Fourier transform. This summation formula will be used to give another proof of the Jacobi transformation formula for the Jacobi theta function. In a forthcoming paper the Mellin summation formula will be of basic importance in studying the Shannon sampling theorem of signal analysis in the Mellin setting, called the exponential sampling theory in optical circles.
We give some Voronovskaja formula for linear combinations of generalized sampling operators and we furnish also a quantitative version in terms of the classical Peetre K-functional. This provides a better order of approximation in the asymptotic formula. We apply the general theory to various kernels: the Bochner-Riesz kernel, translates of B-splines and the Jackson type kernel.
We introduce a class of bivariate discrete operators, not necessarily positive, and we give a general asymptotic formula for the pointwise convergence. Applications to bivariate generalized sampling series and to some Szász-Mirak'jan type operators are given.
We study a class of bivariate generalized sampling operators and we give a general asymptotic formula for the pointwise convergence. Moreover we study a quantitative version.