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Pointwise convergence of generalized Kantorovich exponential sampling series
... where EK w are known with the name of exponential Kantorovich sampling series (see [3,1,2]), and finally: ...
... Definition 8. 1 We define the nonlinear exponential sampling operators for a given nonlinear kernel χ, as follows ...
... Proof 8.1 We begin proving (i). For every fixed x ∈ R + , we can write what follows: ...
... by replacing the value f (e k w ) with the mean value of f (e x ) in the interval [ k w , k+1 w ] for k ∈ Z, w > 0 . Recently, several modifications of it were studied in [2], [3], [5], [6], and [32]. Durrmeyer type exponential sampling series were presented in [10] in the following form ...
... It is important that the generalized exponential sampling series in (1) and Kantorovichtype exponential sampling series in (2) are included by the Durrmeyer-type exponential sampling series in (3). This study aims to show that the Durrmeyer-type exponential sampling series offers us approximations for functions on R + which are not only continuous or uniformly continuous, but also discontinuous such as elements of (Mellin) Orlicz spaces. ...
In this study, by using the concept of modular convergence with the help of a suitable modular functional we obtain main theorem for the (Mellin) Orlicz spaces X η 0 = L φ μ (R +) whose functions don't have to be bounded or continuous. Then we customize our theorems for L p μ (R +)-space and L η α,β μ (R +) using these results. Finally, examples with graphical representations are given for some Durrmeyer type exponential sampling series with special kernels.
... Voronovskaya-type formulas are asymptotic relations for the pointwise approximation by operators of functions in terms of derivatives (see [1,4,5,6,13,27,32,43] for the Voronovskaya formulas of some operators in approximation theory). For the Hermite type sampling operator G n,w we prove the following result. ...
... Concurrently, significant progress has been made in the study of approximation results by means of exponential-type operators (see, e.g., [1][2][3][4][5][6][7]12]). In this respect, Bardaro, Faina, and Mantellini [9] introduced a new family of sampling-type operators, known as exponential sampling series. ...
In this paper, we introduce the nonlinear exponential Kantorovich sampling series. We establish pointwise and uniform convergence properties and a nonlinear asymptotic formula of the Voronovskaja-type given in terms of the limsup. Furthermore, we extend these convergence results to Mellin-Orlicz spaces with respect to the logarithmic (Haar) measure. Quantitative results are also given, using the log-modulus of continuity and the log-modulus of smoothness, respectively, for log-uniformly continuous functions and for functions in Mellin-Orlicz spaces. Consequently, the qualitative order of convergence can be obtained in case of functions belonging to suitable Lipschitz (log-H\"olderian) classes.
... For m ∈ ℕ , applying the operators ̃ , m, on both sides of (4.1), we obtain and of course applying limit m → ∞ on both sides, we get We end this section by presenting two important results for the newly defined operators: one is a quantitative Voronovskaja-type result and the other one is a Grüss-Voronovskaja-type result. These types of results have been studied extensively in the recent years (See [50][51][52][53]). ...
We propose a Beta-type integral generalization of the Bernstein operators associated with Bézier bases and a shape parameter . First, we study a Korovkin-type result for the proposed operators and then establish their rate of convergence with the help of the modulus of continuity and Peetre’s K-functional. We present a quantitative Voronovskaja type and a Grüss-Voronovskaja type results to study their rate of convergence. We estimate the error for absolutely continuous functions having derivatives of bounded variation. Finally, we provide some graphical examples to illustrate our theoretical results. Relevance of the work:- The generalized operator (1.5) is a powerful tool that can be used to approximate continuous functions, integrable functions, Lipschitz-type functions, and functions with derivatives of bounded variation on the bounded interval [0, 1]. This operator can also be used to solve differential equations and integral equations.
... For m ∈ ℕ , applying the operators ̃ , m, on both sides of (4.1), we obtain and of course applying limit m → ∞ on both sides, we get We end this section by presenting two important results for the newly defined operators: one is a quantitative Voronovskaja-type result and the other one is a Grüss-Voronovskaja-type result. These types of results have been studied extensively in the recent years (See [50][51][52][53]). ...
We propose a Beta-type integral generalization of the Bernstein operators associated with Bézier bases ̃pm,l(x) and a shape parameter . First, we study a Korovkin-type result for the proposed operators and then establish their rate of convergence with the help of the modulus of continuity and Peetre’s K-functional. We present a quantitative Voronovskaja type and a Grüss-Voronovskaja type results to study their rate of convergence. We estimate the error for absolutely continuous functions having derivatives of bounded variation. Finally, we provide some graphical examples to illustrate our theoretical results.
... Similarly, generalized Durrmeyer sampling operators (see, [10,21]) provide a further generalization by integrating polynomial terms. Additional developments, such as exponential sampling series and modifications introduced by Bardaro et al. [12] (for further see, [2,3,28]), 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.
... Bajpeyi and Kumar [28] introduced a novel family of NNOs based on exponential sampling and analyzed their convergence properties. Aral et al. [29] introduced a new family of NNOs by generalizing Kantorovich-type exponential sampling series and quantified the rate of convergence of these operators by using logarithmic modulus of continuity. Alagöz et al. [30] presented sampling Durrmeyer operators and discussed the local and global convergence results for continuous functions in a weighed space. ...
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.
... 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, 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.
... Additionally, in [10][11][12][13][14], some investigations have been performed recently on sampling operators in some different spaces, such as Mellin-Lebesgue, Orlicz, etc. ...
In this study, we give a modification of Mellin convolution-type operators. In this way, we obtain the rate of convergence with the modulus of the continuity of the m m th-order Mellin derivative of function f f , but without the derivative of the operator. Then, we express the Taylor formula including Mellin derivatives with integral remainder. Later, a Voronovskaya-type theorem is proved. In the last part, we state order of approximation of the modified operators, and the obtained results are restated for the Mellin-Gauss-Weierstrass operator.
... 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 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.
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 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.
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.
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 present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gauß, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition–Poisson summation formula, the functional equation of Riemann’s zeta function, as well as the Euler–Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.
The paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited signals. Basic is the fact that by these extensions aliasing error terms must now be added to the bandlimited reconstruction formulae. These errors will be estimated in terms of the distance functional just introduced by the authors for the extensions of basic relations valid for bandlimited functions to larger function spaces. This approach can be regarded as a mathematical foundation of aliasing error analysis of many applications.
It is shown that the Whittaker-Kotel'nikov-Shannon sampling theorem of signal analysis, which plays the central role in this article, as well as (a particular case) of Poisson's summation formula, the general Parseval formula and the reproducing kernel formula, are all equivalent to one another in the case of bandlimited functions. Here equivalent is meant in the sense that each is a corollary of the other. Further, the sampling theorem is equivalent to the Valiron-Tschakaloff sampling formula as well as to the Paley-Wiener theorem of Fourier analysis. An independent proof of the Valiron formula is provided. Many of the equivalences mentioned are new results. Although the above theorems are equivalent amongst themselves, it turns out that not only the sampling theorem but also Poisson's formula are in a certain sense the 'strongest' assertions of the six well-known, basic theorems under discussion.
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.
For pt.I see ibid., vol.7, p.1-20 (1991). The general theory of exponential sampling for the inversion of Mellin-type kernels presented in part I of this paper is applied to some practical inversion problems encountered in laser scattering experiments to determine particle size.
The aim of this paper is to present an approach to the Mellin transform that is fully independent of Laplace or Fourier transform
theory, in a systematic, unified form, containing the basic properties and major results under natural, minimal hypotheses
upon the functions in questions. Cornerstones of the approach are two definitions of the transform, a local and global Mellin
transform, the Mellin translation and convolution structure, in particular approximation-theoretical methods connected with
the Mellin convolution singular integral enabling one to establish the Mellin inversion theory. Of special interest are the
Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the
fundamental theorem of the differential and integral calculus in the Mellin frame. These two operators are different than
those considered thus far and more general. They are of particular importance in solving differential and integral equations.
As applications, the wave equation onℝ
+ × ℝ+ and the heat equation in a semi-infinite rod are considered in detail. The paper is written in part from an historical, survey-type
perspective.
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.
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.
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.
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.
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.
The purpose of this overview paper is to present an approach to Mellin transform theory that is fully independent of Laplace or Fourier transform results, in a unified systematic form, one that contains the transform properties and results under natural, minimal assumptions upon the functions in question. Cornerstones are two definitions of the Mellin transform, a local and a global transform, the Mellin inversion theory, established by approximation theoretical methods connected with the Mellin convolution singular integral of Gauss-Weierstrass-type, and especially the Mellin operators of differentiation and anti-differentiation.
In several papers the authors introduced a self-contained approach (independent of Fourier or Laplace transform theory) to classical Mellin transform theory as well as to a new finite Mellin transform in case the functions in question are absolutely (Lebesgue) integrable. In this paper the matter is considered for functions which are basically square-integrable. The unified and systematic approach presented, which is valid under minimal and natural hypotheses, is applied to sampling analysis, particularly to that connected with Kramer's lemma. An application is a clean approach to exponential sampling theory (of optical circles) for signals which possess certain integral representations, but also for Mellin-bandlimited signals as well as for those which are only approximately Mellin-bandlimited.
The Whittaker–Shannon–Kotel'nikov sampling theorem enables one to reconstruct signals f bandlimited to [−πW,πW] from its sampled values f(k/W), k∈Z, in terms of If f is continuous but not bandlimited, one normally considers limW→∞(SWf)(t) in the supremum-norm, together with aliasing error estimates, expressed in terms of the modulus of continuity of f or its derivatives. Since in practice signals are however often discontinuous, this paper is concerned with the convergence of SWf to f in the Lp(R)-norm for 1<p<∞, the classical modulus of continuity being replaced by the averaged modulus of smoothness τr(f;W−1;Mp(R)). The major theorem enables one to sample any bounded signal f belonging to a certain subspace Λp of Lp(R), the jump discontinuities of which may even form a set of measure zero on R. A corollary gives the counterpart of the approximate sampling theorem, now in the Lp-norm.The averaged modulus, so far only studied for functions defined on a compact interval [a,b], had first to be extended to functions defined on the whole real axis R. Basic tools are the de La Vallée Poussin means and a semi-discrete Hilbert transform.
Here, using Mellin derivatives and a different notion of moment, we state a Voronovskaja approximation formula for a class of Mellin–Fejer type convolution operators. This new approach gives direct and simple applications to various important specific examples.
The article discusses several optical configurations used for signal processing. Electronic-to-optical transducers are outlined, noting fixed window transducers and moving window acousto-optic transducers. Folded spectrum techniques are considered, with reference to wideband RF signal analysis, fetal electroencephalogram analysis, engine vibration analysis, signal buried in noise, and spatial filtering. Various methods for radar signal processing are described, such as phased-array antennas, the optical processing of phased-array data, pulsed Doppler and FM radar systems, a multichannel one-dimensional optical correlator, correlations with long coded waveforms, and Doppler signal processing. Means for noncoherent optical signal processing are noted, including an optical correlator for speech recognition and a noncoherent optical correlator.
Approximation by Durrmeyer type exponential sampling operators. Numerical Functional Analysis and Optimization
- S Bajpeyi
- S K Angamuthu
- I Mantellini
S. Bajpeyi, S. K. Angamuthu, I. Mantellini. Approximation by Durrmeyer type exponential sampling operators. Numerical Functional
Analysis and Optimization, 2022, doi.org/10.1080/01630563.2021.1980798.
The exponential sampling theorem of signal analysis. Atti del Seminario Matematico e Fisico dell
- P L Butzer
- S Jansche
P. L. Butzer, S. Jansche. The exponential sampling theorem of signal analysis. Atti del Seminario Matematico e Fisico dell'Universita di
Modena e Reggio Emilia, 46:99-122, 1998, special issue dedicated to Professor Calogero Vinti.
An introduction to sampling analysis
- P L Butzer
- G Schmeisser
- R L Stens
P. L. Butzer, G. Schmeisser, R. L. Stens. An introduction to sampling analysis. In: Marvasti, F. (ed.) Nonuniform Sampling, Theory and
Practice, Kluwer, New York, 2001, pp. 17-121.