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A characterization of the rate of the simultaneous approximation by generalized sampling operators and their Kantorovich modification
Abstract
We establish a direct and a matching two-term strong converse inequality by moduli of smoothness for the rate of the simultaneous approximation by generalized sampling operators and their Kantorovich modification in the Lp-norm, in particular, the uniform norm on R. They yield the saturation property and class for the simultaneous approximation by the generalized sampling operators, and in the case of the sampling Kantorovich operators---the saturation class of the operator itself as well.
... lim w→+∞ ∥G w f − f ∥ ∞ = 0, if f is bounded and uniformly continuous [38,19]. Generalized sampling operators have been further studied in relation to Voronovskaya-type results [13], approximation of curves [21], simultaneous approximation [3] and have been modied in the multivariate case in [16,17] and for L p functions in [12]. ...
... Reconstruction in shift-invariant spaces with derivative samplings were considered in [24]; the authors gave also approximation results but assuming as additional hypotheses the StrangFix conditions of a certain order on the generator of the space. In [36] the authors have studied the expansions in shift-invariant 3 The name "Hermite-type sampling operator of order n" is also simpler than "generalization of the n-th order of the generalized sampling operator", terminology that could be adopted from [29]. 4 Actually, it is sucient that the series (1.7) is convergent uniformly for u ∈ [0, 1[ since it is periodic in u with period 1. spline spaces by operators involving derivatives and reconstruction functions, which are obtained from the generator of the space and a block Laurent operator. They also studied the approximation under the condition that the operators x the polynomials up to a certain order. ...
... and by the boundedness of f (n) we have Since the Hermite-type sampling operators G n,w , n ≥ 1, are dened for functions with a certain degree of smoothness, it is natural to ask if the derivatives of the G n,w f converge to the derivatives of f . For the generalized sampling operator a result about this topic, i.e. about the simultaneous approximation, has been given in [3]. The following theorem gives an answer to this question. ...
... introduced and deeply studied by Butzer and his school since the 80s (see, e.g., [1,6,13,15,20,41,48]), opening the way for a rigorous study on the generalized sampling series with important applications to the signal and image processing. In particular, by taking the Delta distribution as ψ in (1), we can see that (3) are also a particular case of (1). ...
... In particular, by taking the Delta distribution as ψ in (1), we can see that (3) are also a particular case of (1). However, contrary to (1) and (2), the operators in (3) are no longer bounded in L p -spaces and also the convergence property (in L p -sense) holds only for functions in suitable subspaces (see, e.g., [6]). The chief purpose of this work is to continue the study on the approximation properties of (1) in L p -spaces, focusing now the attention on their regularization properties. ...
... Now, we recall the definition of the family of semi-discrete sampling operators of Durrmeyer-type (1), whose regularization properties will be the aim of the present study. We start from the definition of the kernel functions ϕ and ψ that define (1). From now on, we will say that a function ϕ : R → R is a discrete kernel if the following conditions hold: ...
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.
... Following this field of research, in this paper we introduced a new class of samplingtype operators, named Steklov sampling operators S r w (SSO r , see Sect. 3). ...
... Among the main studied sampling-type series we can mention the following: the generalized sampling operators G χ w (see, e.g, [3,13,15]), the Kantorovich sampling operators K χ w (see, e.g, [10,16,17]), and finally the Durrmeyer sampling operators D χ,ψ w (see, e.g, [9,23,35,36,38,39]). We briefly recall their definitions, which are the following: ...
... In this case, if we consider, e.g., functions f belonging to C 2 (R) we find the above described difference concerning the order of approximation between S r w f , r ≥ 2 and K χ w f (see [25] again) with respect to the uniform norm. Hence, the Steklov sampling 3 Note that, in the computation of the A χ j it is crucial to have at disposal the j-th derivatives of the Fourier transform of χ at the nodes 2π k, k ∈ Z. This fact is a consequence of the well-known Poisson summation formula; for more details see, e.g., [12,26]. ...
In this paper we introduce a new class of sampling-type operators, named Steklov sampling operators. The idea is to consider a sampling series based on a kernel function that is a discrete approximate identity, and which constitutes a reconstruction process of a given signal f , based on a family of sample values which are Steklov integrals of order r evaluated at the nodes k / w , k ∈ Z , w > 0 . The convergence properties of the introduced sampling operators in continuous functions spaces and in the L p -setting have been studied. Moreover, the main properties of the Steklov-type functions have been exploited in order to establish results concerning the high order of approximation. Such results have been obtained in a quantitative version thanks to the use of the well-known modulus of smoothness of the approximated functions, and assuming suitable Strang-Fix type conditions, which are very typical assumptions in applications involving Fourier and Harmonic analysis. Concerning the quantitative estimates, we proposed two different approaches; the first one holds in the case of Steklov sampling operators defined with kernels with compact support, its proof is substantially based on the application of the generalized Minkowski inequality, and it is valid with respect to the p -norm, with 1 ≤ p ≤ + ∞ . In the second case, the restriction on the support of the kernel is removed and the corresponding estimates are valid only for 1 < p ≤ + ∞ . Here, the key point of the proof is the application of the well-known Hardy–Littlewood maximal inequality. Finally, a deep comparison between the proposed Steklov sampling series and the already existing sampling-type operators has been given, in order to show the effectiveness of the proposed constructive method of approximation. Examples of kernel functions satisfying the required assumptions have been provided.
... Among these, we recall the well-known generalized sampling type operators ( [18][19][20]), that arise from a distributional version of Durrmeyer series where ψ is the Dirac delta distribution. Furthermore, taking as ψ the characteristic function of the interval [0, 1], we reduce to the celebrated Kantorovich sampling type operators ( [1][2][3]6,21,27,30]). In our earlier papers, we approach the problem of convergence for Durrmeyer sampling type operators in the general framework of Orlicz spaces ( [14,15]), both in one and in more variables (see, respectively, [24,25] and [23]), including the L p -spaces as particular case. ...
... In this case, we point out that m 1 (χ [0,1] ) = 1 2 ̸ = 0, so that the vanishing moment condition (ii) is not satisfied for r = 2. In general, it is easy to see that m ν (χ [0,1] ) = 1 ν+1 ̸ = 0, for every ν ∈ N. Hence, in order to improve the order of approximation up to 2, we should replace χ [0, 1] with a symmetric characteristic functions of the form χ [−a,a] 2a , with a > 0, obtaining again operators of Kantorovich type. In fact, in this case all the continuous moments of order odd vanish. ...
In this paper, we establish a comprehensive characterization of the generalized Lipschitz classes through the study of the rate of convergence of a family of semi-discrete sampling operators, of Durrmeyer type, in -setting. To achieve this goal, we provide direct approximation results, which lead to quantitative estimates based on suitable K-functionals in Sobolev spaces and, consequently, on higher-order moduli of smoothness. Additionally, we introduce a further approach employing the celebrated Hardy-Littlewood maximal inequality to weaken the assumptions required on the kernel functions. These direct theorems are essential for obtaining qualitative approximation results in suitable Lipschitz and generalized Lipschitz classes, as they also provide conditions for studying the rate of convergence when functions belonging to Sobolev spaces are considered. The converse implication is, in general, delicate, and actually consists in addressing an inverse approximation problem allowing to deduce regularity properties of a function from a given rate of convergence. Thus, through both direct and inverse results, we establish the desired characterization of the considered Lipschitz classes based on the -convergence rate of Durrmeyer sampling operators. Finally, we provide remarkable applications of the theory, based on suitable combinations of kernels that satisfy the crucial Strang-Fix type condition used here allowing to both enhance the rate of convergence and to predict the signals.
... This theorem and its various extensions and generalizations have been proved in many different ways, for example, using Fourier expansion, the Poisson summation formula, contour integrals, and so on (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein). For instance, in [12] the sampling theorem (1.1) was established via Cauchy's residue theorem for the entire functions which satisfy an inequality ...
In this paper, we prove that the sampling theorem for the product of two entire functions, which grows no faster than reciprocal gamma functions, implies the generalization of the Lerche–Newberger formula for the series involving product of Bessel functions of the first kind. As a consequence of the generalized Lerche–Newberger formula, we present a new addition formula for the Bessel function, a particular case of which extend the well‐known Graf 's addition formula.
... 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 ≤ ∞. Angeloni et al. [34] proved a quantitative estimate for multivariate sampling Kantorovich operators in terms of modulus of smoothness in Orlicz spaces and showed that for L p spaces, a more precise estimate of the convergence rate can be obtained compared with the general case of Orlicz spaces. ...
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.
... Moreover, the S w operator can also be used to reconstruct not-necessarily continuous signals, e.g., signals belonging to the L p -spaces, 1 ≤ p < +∞ ( [2-5, 8-10, 12, 13, 19, 29, 30, 33, 40, 43, 44, 52]). The function χ : R 2 → R, given in (3.5), is called a kernel and it satisfies the following suitable assumptions, very typical in this situation, which are the usual conditions assumed by discrete approximate identities (for more details, see, e.g., [1]). Below, we present a list of functions that can be used as kernels in the formula recalled in (3.5). ...
The aim of this paper is to compare the fuzzy-type algorithm for image rescaling introduced by Jurio et al., 2011, quoted in the list of references, with some other existing algorithms such as the classical bicubic algorithm and the sampling Kantorovich (SK) one. Note that the SK algorithm is a recent tool for image rescaling and enhancement that has been revealed to be useful in several applications to real world problems, while the bicubic algorithm is widely known in the literature. A comparison among the abovementioned algorithms (all implemented in the MatLab programming language) was performed in terms of suitable similarity indices such as the Peak-Signal-to-Noise-Ratio (PSNR) and the likelihood index S.
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.
We establish a direct and a matching two-term converse estimate by a K-functional and a modulus of smoothness for the rate of approximation by generalized Kantorovich sampling operators in variable exponent Lebesgue spaces. They yield the saturation property and class of these operators. We also prove a Voronovskaya-type estimate.
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.
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 the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in L p -setting, 1 ≤ p ≤ + ∞ , have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel influences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order r ∈ N + . Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the L p modulus of smoothness of order r have been established. As a consequence, the qualitative order of approximation is also derived assuming f in suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncovered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in L p ( R ) , 1 ≤ p ≤ + ∞ , for the sampling Kantorovich operators.
Approximation properties of quasi-projection operators Q j ( f , φ , φ ~ ) are studied. These operators are associated with a function φ satisfying the Strang–Fix conditions and a tempered distribution φ ~ such that compatibility conditions with φ hold. Error estimates in the uniform norm are obtained for a wide class of quasi-projection operators defined on the space of uniformly continuous functions and on the anisotropic Besov-type spaces. Under additional assumptions on φ and φ ~ , two-sided estimates in terms of realizations of the K -functional are also established.
In this paper, we discuss various basic properties of moduli of smoothness of functions from Lp(Rd), 0<p≤∞. In particular, complete versions of Jackson-, Marchaud-, and Ulyanov-type inequalities are given for the whole range of p. Moreover, equivalences between moduli of smoothness and the corresponding K-functionals and the realization concept are proved.
The study of inverse results of approximation for the family of sampling Kantorovich operators in case of α-Hölder function, 0 < α < 1, has been solved in a recent paper of some of the authors. However, the limit case of Lipschitz functions, i.e., when α = 1, in which standard methods fail, remained unsolved. In this paper, a solution of the above open problem of inverse approximation has been proposed.
Approximation properties of the sampling‐type quasi‐projection operators Qj(f,φ,φ˜) for functions f from anisotropic Besov spaces are studied. Error estimates in Lp‐norm are obtained for a large class of tempered distributions φ˜ and a large class of functions φ under the assumptions that φ has enough decay, satisfies the Strang‐Fix conditions, and has a compatibility condition with φ˜. The estimates are given in terms of moduli of smoothness and best approximations.
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 extend the saturation results for the sampling Kantorovich operators proved in a previous paper, to more general settings. In particular, exploiting certain Voronovskaja-formulas for the well-known generalized sampling series, we are able to extend a previous result from the space of -functions to the space of -functions. Further, requiring an additional assumption, we are able to reach a saturation result even in the space of the uniformly continuous and bounded functions. In both the above cases, the assumptions required on the kernels, which define the sampling Kantorovich operators, have been weakened with respect to those assumed previously. On this respect, some examples have been discussed at the end of the paper.
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.
This paper presents methods for the reconstruction of band-limited functions from irregular samples. The first part discusses algorithms for a constructive solution of the irregular sampling problem. An important aspect of these algorithms is the explicit knowledge of the constants involved and efficient error estimates. The second part discusses the numerical implementation of these algorithms and compares the performance of various reconstruction methods. Although most of the material has already appeared in print, several new results are included: (a) a new method to estimate frame bounds; (b) a reconstruction of band-limited functions from partial information, which is not just the samples; (c) a new result on the complete reconstruction of band-limited functions from local averages; (d) a systematic exposition of recent experimental results.” Contents: 8.1. Introduction: The irregular sampling problem. I. A quantitative theory of irregular sampling: 8.2. Iterative reconstruction algorithms; 8.3. Qualitative sampling theory; 8.4. The adaptive weights method; 8.5. Variations; 8.6. Irregular sampling of the wavelet and the short time Fourier transform. II The practice of irregular sampling: 8.7. Setup for numerical calculations; 8.8. Hard and software environment: PC386/486 and Sun, IRSATOL/MATLAB; 8.9. Results and observations from the experiments; 8.10. Additional results and observations; 8.11. Conclusion; Bibliography.
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 [10, 11, 12] we introduced a new family of algorithms for the reconstruction of a band-limited function from its irregularly sampled values. In this paper we carry out an error analysis of these algorithms and discuss their numerical stability. As special cases we also obtain more precise error estimates in the case of the regular sampling theorem. 1 Introduction This paper is devoted to the error analysis of irregular sampling theorems that were developed in [10, 11, 12]. It is a direct continuation of our previous work, the separation is only for reasons of length and tradition. The preceding articles [10, 11, 12] present a new approach to the irregular sampling problem for band-limited functions. They were motivated by the demand for efficient algorithms for the reconstruction of a band-limited function from its irregularly sampled values, a problem that occurs frequently in signal and image processing, geophysics and other applications. Using real-variable methods only, we found...
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.
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.
Multivariate quasi-projection operators Qj(f,φ,φ˜), associated with a function φ and a distribution/function φ˜, are considered. The function φ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with φ˜. Using technique based on the Fourier multipliers, we study approximation properties of such operators for functions f from anisotropic Besov spaces and Lp spaces with 1≤p≤∞. In particular, upper and lower estimates of the Lp-error of approximation in terms of anisotropic moduli of smoothness and anisotropic best approximations are obtained.
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 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.
Approximation properties of multivariate Kantorovich-Kotelnikov type operators generated by different band-limited functions are studied. In particular, a wide class of functions with discontinuous Fourier transform is considered. The -rate of convergence for these operators is given in terms of the classical moduli of smoothness. Several examples of the Kantorovich-Kotelnikov operators generated by the sinc-function and its linear combinations are provided.
A constructive solution of the irregular sampling problem for bandlimited functions is given. We show how a band-limited function can be completely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions.
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.
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.
We estimate the L(p)(R(d))_approximation rate (1 less-than-or-equal-to p less-than-or-equal-to infinity) provided dilates of an orthogonal projection operator of L2(R(d)) onto a space generated by shifts of a function which has a polynomial decay rate at infinity and has stable shifts. To this end we employ a quasi-interpolation scheme and invoke Wiener's lemma. In particular, this paper provides further substantiation to one of the fundamental properties of refinable functions: the connection between smoothness and approximation order. (C) 1994 Academic Press, Inc.
Techniques are developed to obtain strong converse inequalities for various linear approximation processes. This will establish
equivalence between the approximating rate of a certain linear process and the appropriate PeetreK-functional. Approximation processes that will be treated have to be saturated asK-functionals are saturated. These general methods will lead to new results on the various trigonometric polynomial approximation
processes, on holomorphic semigroups, on Bernstein polynomials and on other commonly used approximation processes.
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.
The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.
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.
The lower estimate for linear positive operators (II)
- Knoop
On generalized sampling sums based on convolution integrals
- Splettstosser
Error analysis in regular and irregular sampling theory
- Butzer