Search

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 study boundedness properties of certain semi-discrete sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed. These results pave the way to the norm-convergence of these operators

Let $B^p_{\sigma}$, $1\le p0$, denote the space of all $f\in L^p(\mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-\sigma,\sigma]$. The classical sampling theorem states that each $f\in B^p_{\sigma}$ may be reconstructed exactly from its sample values at equispaced sampling points $\{\pi m/\sigma\}_{m\in\mathbb{Z}} $ spaced by $\pi /\sigma$. Reconstruction is also possible from sample values at sampling points $\{\pi \theta m/\sigma\}_m $ with certain $1< \theta\le 2$ if we know $f(\theta\pi m/\sigma) $ and $f'(\theta\pi m/\sigma)$, $m\in\mathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives., Comment: 10 pages

We investigate the Gibbs–Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions. We prove the existence of the overshoot characteristic of the phenomenon for certain cardinal functions, and characterize the existence of an overshoot for sampling series.

We study sampling series and their relationship to frequency band limited functions. Motivated by the theories of multiresolution analyses and subdivision, particular attention is paid to sampling series whose kernels are refinable.

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.

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 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.

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, 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 out 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 Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

Growth conditions are given on the samples f(n), n = 0, ±1, ±2,…, of an entire function f(z) of exponential type less than π that imply that the corresponding cardinal sine series converges. These conditions are the least restrictive of their kind that are possible. Furthermore, an example is provided of an entire function f(z) of exponential type π that is bounded on the real axis and whose corresponding cardinal sine series fails to converge.

This article discusses the structure of various shift-invariant spaces of cardinal functions generated by a single kernel, and their role in some approximate sampling methods. Particularly, conditions on the generating kernel φ are given which imply that the associated L p shift-invariant space coincides with the shift-invariant space for the cardinal function L φ generated by the kernel. Additionally, we discuss how such spaces are used in summability methods for the classical sampling series.

Many families of cardinal sampling kernels { Φ α ( x ) : α ∈ I } consist of members that decay rapidly as x → ± ∞ and enjoy the property that, as the parameter α tends to its limiting value, Φ α ( x ) tends to the classical cardinal sine, sinc ( x ) . We study the limiting behavior of the sampling series ∑ n c n Φ α ( x − n ) when the data sequence { c n } remains fixed. In the case when the sampling kernels are damped cardinal sines, appropriate conditions on the damping functions allow us to characterize the limiting behavior of the corresponding sampling series for a wide class of data sequences { c n } . These results lead to conclusions regarding the limiting behavior of classical piecewise polynomial cardinal splines when the fixed data sequence { c n } is of polynomial growth and the order tends to ∞ . One application of this development shows that the classical spline summability method is regular. Another consequence, perhaps more interesting and unexpected, is a resolution of I. J. Schoenberg’s conjecture concerning the recovery, in terms of their samples { f ( n ) } , of members f ( x ) of the Bernstein class via the spline summability procedure.

Errors appear when the Shannon sampling series is applied to approximate a signal in practice since the measured sampled values are usually given by averages of a function or the sampled values with amplitude error and time-jitter error. In this paper we give a survey of recent progress of the study of error analysismulti-dimensional Whittaker-Shannon sampling expansion approximations with these measured values.

Elements of Bernstein spaces of band-limited functions with band width σ, σ > 0 are perfectly sampled from their values at discrete set of points with different convergence criteria. The aliasing phenomenon occurs if the function is not band-limited or the sampling rate is lower than the band-width Nyquist rate), i.e σ → ∞. Since both conditions, band-limitedness and Nyquist rate are restrictive, it is desirable to find rigorous error estimates for the aliasing error. Here, several bounds for the aliasing error of sampling derivatives are established. We derive both uniform and Lp-norm bounds which are analogues of the results of Butzer et al. (2005), Fang (1996). Moreover, the so-called truncated aliasing error is investigated and a few numerical examples will be presented.

We study the problem of approximating a real-valued function f by considering sequences of general operators of sampling type, which include both discrete and integral ones. This approach gives a unitary treatment of various kinds of classical operators, such as Urysohn integral operators, in particular convolution integrals, and generalized sampling series.

We consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method, called the Gaussian regularization of the Shannon series, was proposed in the engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. Recently, it was proved that the upper error bound of this method can achieve a convergence rate of the order $O(\frac{1}{\sqrt{n}}\exp(-\frac{\pi-\delta}{2}n))$, where $0, Comment: 14 pages, 1 figure

This chapter consists of an overview of recent results concerning the convergence and regularization of the quintessential sampling series, the cardinal sine series. Conditions, that go beyond those associated with the standard theory, are formulated that ensure reconstruction by this series. The conditions are one of two types: (i) on the coefficients or samples or (ii) on the functions or signals being reconstructed. A class of regularization methods, that in effect consist of mollifications of cardinal sine series, is discussed. One of the highlights is a result that shows how piecewise polynomial splines and certain variants can be regarded as such mollifications.

Errors appear when the Shannon sampling series is applied to reconstruct a signal in practice. In this paper, we study a general model that uses linear functional to cover several errors in one formula, and consider sampling series with measured sampled values for not band-limited signals but satisfying some decay condition. We obtain the uniform truncated error bound of Shannon series approximation for multivariate Besov class. The results show this kind of Shannon series can approximate a smooth signal well.

In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We deduce a sampling series in terms of these functions for Hankel–band–limited signals and derive bounds for the truncation error of the sampling series.

For any ϕ(t) in L2(ℝ), let V(ϕ) be the closed shift invariant subspace of L2(ℝ) spanned by integer translates {ϕ(t - n) : n ∈ ℤ} of ϕ(t). Assuming that ϕ(t) is a frame or a Riesz generator of V(ϕ), we first find conditions under which V(ϕ) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(ϕ) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L2(ℝ) that belongs to a sampling subspace of L2(ℝ). Several illustrating examples are also provided.

Alising error arises whenever a sampling formula, valid for a prescribed space, is applied to a function in a bigger space. In this work, we estimate the aliasing error of classic and average sampling expansions in wavelet subspaces of a multiresolution analysis.

Localization operator based on sampling multipliers is proposed to reconstruct a function in Paley–Wiener spaces by its local samples. Explicit error estimate is derived for the operator to approximate functions and their derivatives. For Hermite multipliers, exponentially decaying accuracy of approximation is achieved, and a practical criteria for the sampling to obtain any desired accuracy is provided. The estimates unify several existing results for sampling and accelerate the convergence rate of wavelet sampling series.

In this paper, the behavior of the sampling Kantorovich operators has been studied, when discontinuous 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. Further, also the problem of the linear prediction by sampling values from the past is analyzed. At the end, the role of duration-limited kernels in the previous approximation processes has been treated, and several examples have been provided., 22 pages

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(SWf)(t)≡∑k=−∞∞f(kW)sinc(Wt−k)=f(t)(t∈R). 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

We propose a modification of the sampling series using a Gaussian multiplier. Error estimates for the modified series to approximate a band-limited function and its derivatives are obtained. The error bounds decay exponentially. A criterion for choosing the parameters to achieve a desired accuracy is provided. Numerical experiments show that the Gaussian multiplier performs better than the Jagerman multiplier for functions with high bandwidth.

We propose generalized sinc kernels for the localized sampling series, derive explicit error estimates for the sampling series to approximate a band-limited function and its derivatives, and provide rigorous and practical means for obtaining any desired accuracy for sampling. Theoretically and numerically, we demonstrate that the approximation is highly efficient.

Let (H,〈ċċ〉 H ) be a complex, separable Hilbert space with orthonormal basis {x n } n=1 ∞ and let Ω be a domain in C, the field of complex numbers. Suppose K is a H-valued function defined on Ω. For each x ∈ H, define f x (z)= 〈K(z),x〉H and let H denote the collection of all such functions f x . In this paper, we endow H with a structure of a reproducing kernel Hilbert space. Furthermore, we show that each element in H is analytic on Ω if and only if K is analytic on Ω or, equivalently, if and only if 〈K(z),x n 〉 is analytic for each n ∈ N and ||K(ċ)||H is bounded on all compact subsets of Ω. In this setting, an abstract version of the analytic Kramer theorem is exhibited. Some examples considering different H spaces are given to illustrate these new results.

Error estimation is given for the regularized Whittaker-Kotel'nikov-Shannon (WKS) sampling formula, which was found to be accurate and robust for numerically solving partial differential equations. The result improves the convergence rate of existing results.

We give some new estimates for the Truncation Error of sampling series of functions on regular sampling subspaces of . These estimates lower the well known Jagerman's bound on Shannon's sampling expansions.

This article is devoted to the derivation of sampling series associated with eigenvalue problems. No examples of sampling theorems associated with odd order problems are known except when the order is one. Here we give necessary and sufficient conditions for the boundary conditions to be self adjoint. Then sampling series associated with odd order self adjoint and non self adjoint problems are given. The sampling representations associated with non self adjoint odd order problems are derived provided that the boundary condition are regular.

The sampling theory says that a function may be determined by its sampled values at some certain points provided the function satisfies some certain conditions. In this paper, we consider the convergence of the sampling series for bounded functions with the sampling function from some multiresolution analysis. We also present a convergence theorem on the sampling series with irregular sampling points and give the approximation error.

Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.

In this paper we investigate the behavior of the aliasing error of non bandlimited sampled signals. The Shannon sampling series have an optimal reconstruction behavior for bandlimited signals. In the case of non bandlimited signals the Shannon sampling series have a complete different behavior. For these signals the Shannon sampling series dont approximate the sampled signals in general case.

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous function g1∈C0 [0,1]2 with support in the rectangle [0,1] × [0,½] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1] × [½,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer.

Sampling series, where the classical sinc function is replaced by the general kernel function, are studied widely by P.L. Butzer and his school at the RWTH Aachen University since 1977. To study L p -convergence they introduced recently a certain class \(\varLambda ^{p} \subset L^{p}(\mathbb{R})\), and the rate of approximation was estimated via the τ-modulus.

In this paper some generalized Shannon sampling operators which diminish the total variation of functions and their derivatives are considered. For that purpose will be used some related kernel functions of certain even bandlimited kernel functions.

Errors appear when the Shannon sampling series is applied to reconstruct a signal in practice. This is because the sampled values may not be exact, or the sampling series may have to be truncated. In this paper, we study errors in truncated sampling series with localized sampling for band-limited functions from weighted Bernstein class. And we apply these results to some practical examples. DOI: http://dx.doi.org/10.11591/telkomnika.v11i8.3078

Let {Vm be a multiresolution analysis of L2(R) such that a sampling function S for V0 exists. Then we show the sampling approximation of a function in H0,α1/2 onto Vm converges to it uniformly as m→∞. Also Gibbs phenomenon for this sampling expension is analyzed.

In this paper we derive rates of approximation for a class of linear operators on \(C_B({\bf R}^d)\) associated with a multiresolution analysis \(\{ V_n\}_{n\in{\bf Z}}.\) We show that for a uniformly bounded sequence of linear operators \(\{T_n\}_{n\in{\bf Z}}\) satisfying \(T_n f\equiv f\) on the subspace \(V_n\cap C_B({\bf R}^d),\) a lower bound for the approximation order is determined by the number of vanishing moments of a prewavelet set. We consider applications to extensions of generalized projection operators as well as to sampling series.

It is shown that bandlimited Lp-functions can be reconstructed by sampling series of Lagrange type with knots {tn}n∈Z whenever |tn−n|≤L

Motivated by a recent letter on an infinite difference between the peak-to-average power ratios (PAPRs) of samples (a series) and a band-limited function constructed from that series, we investigate the amplitude and variations of a band-limited function and present some relevant bounds. Related aspects on sampling theorems and sampling series are also discussed.

The famous Shannon-Whittaker sampling theorem states that if ƒ(t) = ∝−ππe−itu F(u) du, F ϵ L2 [−π, π] then ƒ can be represented by a sampling sum as f(t)=∑n=∘∘f(n)sinπ(t−n)π(t−n)It is shown here that certain functions of polynomial growth can be approximated by a generalized sampling sum and a rate of convergence of the sum to the function is given.