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Convergence properties of Durrmeyer-type sampling operators
Abstract
In this article, we study the approximation properties of Durrmeyer-type sampling operators. We consider the composition of generalized sampling operators and Durrmeyer sampling operators. For the new composition operators, we provide the pointwise and uniform convergence, as well as the quantitative estimates in terms of the first-order modulus of continuity and K-functional. Moreover, we investigate the rate of convergence using weighted modulus of continuity. Also, difference estimates are provided for the operators. Additionally, we provide approximation results for the linear combinations of the composition operators. Finally, we discuss the rate of convergence for the operators via graphical example.
In the present paper, we study the approximation properties of the compositions of Jain operators and Szász–Mirakjan operators. First, we estimate the moment-generating function and moments of the new operators in terms of the Lambert W function and then we establish some convergence results and their quantitative estimates for the difference of the operators, while emphasizing on the preservation of the modulus of continuity. We further discuss Korovkin-type theorem using a Chebyshev system of exponential functions and Voronovskaja-type asymptotic formulae for these operators. In the last section, we provide a comparative study of the rates of convergence of these operators using graphs and numerical tables.
In this article, we capture a new Durrmeyer type operator, linking adjoint Bernoulli polynomials with the Baskakov basis functions. We estimate its moments and central moments and provide approximation results which include Korovkin-type estimate in weighted space. Additionally, we present a graphical interpretation of the operators applied to certain functions, quantitative Voronovskaja’s theorem in weighted space, and a theorem involving Peetre’s K-functional. Furthermore, we give theorems based on difference of operators.
The present paper provides the study on the composition operators due to Szász-Mirakyan with the generalized Szász-Durrmeyer operators. We get an interesting new operator based on Touchard polynomials. Also, the new operator can be further decomposed into two different operators. This study is extension of the recent work, where we have discussed such compositions with reverse order. We find some convergence behavior of these composition operators in sense of point-wise estimates and quantitative estimates in terms of modulus of continuity. We also provide some graphs to have convergence visualization for different n.
In this paper, we are concerned with the study of the regularization properties of Durrmeyer-sampling type operators in -spaces, with . In order to reach the above results, we mainly use tools belonging to distribution theory and Fourier analysis. Here, we show how the regularization process performed by the operators is strongly influenced by the regularity of the discrete kernel . We investigate the classical case of continuous kernels, the more general case of kernels in Sobolev spaces, as well as the remarkable case of bandlimited kernels, i.e., belonging to Bernstein classes. In the latter case, we also establish a closed form for the distributional Fourier transform of the above operators applied to bandlimited functions. Finally, the main results presented herein will be also applied to specific instances of bandlimited kernels, such as de la Vallée Poussin and Bochner–Riesz kernels.
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.
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.
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.
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.
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 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).
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 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.
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.
We study the order of approximation in modular spaces for a family of nonlinear integral operators of the form (T ω f)(s)=∫ H K ω (s-h ω (t),f(h ω (t)))dμ H (t),ω>0,s∈G, where G and H are locally compact topological groups, fG→ℝ is measurable, {h ω } ω>0 is a family of homeomorphisms h ω H→h ω (H)⊂G and {K ω } ω>0 is a family of kernel functions. The general setting of modular spaces allows us to obtain, in particular, the rate of approximation for the above operators in L p spaces and in Orlicz-type spaces. Furthermore, the above general class contains, as particular cases, some classical families of integral operators well known in approximation theory, such as the classical convolution integral operators, the Mellin convolution integral operators and the sampling-type operators in their nonlinear form. Our approach, in the framework of modular spaces, is mainly based on the introduction of a suitable Lipschitz class and of a condition on a family of measures which is linked with the modulars involved and which is always fulfilled in classical and Musielak-Orlicz spaces.
In this article, we study a nonlinear 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-phi((n)) that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in L-p((n))-spaces, LlogL((n))-spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and image processing applications are included.
http://www.tandfonline.com/doi/abs/10.1080/01630563.2013.767833#.Ut0JcPtd7Gg
Shannon’s sampling theorem is one of the most powerful results in signal analysis. The aim of this overview is to show that one of its roots is a basic paper of de la Vallée Poussin of 1908. The historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions (which includes the duration-limited case actually studied in 1908), is sketched. Emphasis is put on the study of error estimates, as well as on the delicate pointwise behavior of sampling sums at discontinuity points of the signal to be reconstructed.
In this paper we introduce a nonlinear version of the Kantorovich sampling type series in a nonuniform setting. By means of the above series we are able to reconstruct signals (functions) which are continuous or uniformly continuous. Moreover, we study the problem of the convergence in the setting of Orlicz spaces: this allows us to treat signals which are not necessarily continuous. Our theory applies to Lp-spaces, interpolation spaces, exponential spaces and many others. Several graphical examples are provided.
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 this paper we study the performance of the sampling Kantorovich (S–K) algorithm for image processing with other well-known interpolation and quasi-interpolation methods. The S-K algorithm has been implemented with three different families of kernels: central B-splines, Jackson type and Bochner–Riesz. The above method is compared, in term of PSNR (Peak Signal-to-Noise Ratio) and CPU time, with the bilinear and bicubic interpolation, the quasi FIR (Finite Impulse Response) and quasi IIR (Infinite Impulse Response) approximation. Experimental results show better performance of S-K algorithm than the considered other ones.
In this paper we study the max-product version of the generalized sampling operators based upon a general kernel function. In particular, we prove pointwise and uniform convergence for the above operators, together with a certain quantitative Jackson-type estimate based on the first order modulus of continuity of the function being approximated. The proof of the proposed results are based on the definition of the so-called generalized absolute moments. By the proposed approach, the achieved approximation results can be applied for several type of kernels, not necessarily duration-limited, such as the sinc-function, the Fejér kernel and many others. Examples of kernels with compact support for which the above theory holds can be given, for example, by the well-known central B-splines.
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
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.
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.
Here we give a Voronovskaya type formula for Kantorovich gen-eralized sampling series and a corresponding quantitative version in terms of some moduli of smoothness.
The sinc-kernel function of the sampling series is replaced by spline functions having compact support, all built up from the B-splines M//n. The resulting generalized sampling series reduces to finite sums so that no truncation error occurs. Moreover, the approximation error generally decreases more rapidly than for for the classical series when W tends to infinity, 1/W being the distance between the sampling points. For the kernel function PHI (t) equals 5M//4(t) minus 4M//5(t) with support left bracket minus 5/2, 5/2 right bracket , e. g. , it decreases with order O(W** minus **4) provided the signal has a fourth order derivative; for the classical series the order is O(W** minus **4 log W). Seven characteristic examples are treated in detail.
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.
Kramer's generalization of Shannon's sampling theorem takes us from a signal represented by a finite Fourier transform to a signal represented by another and more general finite integral transform. In this paper we will attempt to show that the already obtained results for Kramer's theorem are of use in the field of finite integral transforms. Also by introducing such transforms one can treat some communications problems. An example is the case of representing a signal which is the output of time variant filter.
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.
In contrast to the classical Shannon sampling theorem, signal functions are considered which are not band-limited but duration-limited. It is shown that these functions can be approximately represented by a discrete set of samples. The error is estimated that arises when only a finite number of samples is selected.
Using a set of low resolution images it is possible to reconstruct high resolution information by merging low resolution data on a finer grid. A 3D super-resolution algorithm is proposed, based on a probabilistic interpretation of the n-dimensional version of Papoulis' (1977) generalized sampling theorem. The algorithm is devised for recovering the albedo and the height map of a Lambertian surface in a Bayesian framework, using Markov random fields for modeling the a priori knowledge
A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maxmum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of "ideal" systems which transmit at this maxmum rate are discussed. The equivalent number of binary digits per second for certain information sources is calculated.
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An introduction to information theory
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- FM Reza