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In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function ρ which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions. [ABSTRACT FROM AUTHOR]

In this study, we prove the convergence theorems on the space of compactly supported functions and in the general setting of Orlicz spaces for a general class of Kantorovich type discrete operators defined by Carlo Bardaro and Ilaria Mantellini. We also define the generalized Boolean sum (GBS) operator for the class of bivariate Kantorovich type discrete operators and examine the approximation properties of GBS operators in the space of Bögel functions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The sinc-Gaussian sampling formula is used to approximate an analytic function, which satisfies a growth condition, using only finite samples of the function. The error of the sinc-Gaussian sampling formula decreases exponentially with respect to N, i.e., N− 1/2e−αN, where α is a positive number. In this paper, we extend this formula to allow the approximation of derivatives of any order of a function from two classes of analytic functions using only finite samples of the function itself. The theoretical error analysis is established based on a complex analytic approach; the convergence rate is also of exponential type. The estimate of Tanaka et al. (Jpan J. Ind. Appl. Math. 25, 209–231 2008) can be derived from ours as an immediate corollary. Various illustrative examples are presented, which show a good agreement with our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we study the spaceability of divergence sets of sequences of bounded linear operators on Banach spaces. For Banach spaces with the s-property, we can give a sufficient condition that guarantees the unbounded divergence on a set that contains an infinite dimensional closed subspace after the zero element has been added. This generalizes the classical Banach-Steinhaus theorem which implies that the divergence set is a residual set. We further prove that many important spaces, e.g., ℓp, 1≤p<∞, C[0, 1], Lp, 1

Recently Fang and Li established a sampling formula that involves only samples from the function and its first partial derivatives for functions from Bernstein space, Bσp(ℝ²) . In this paper, we derive a general bivariate sampling series for the entire function of two variables that satisfy certain growth conditions. This general bivariate sampling formula involves samples from the function and its mixed and nonmixed partial derivatives. Some known sampling series will be special cases of our formula, like the sampling series of Parzen, Peterson and Middleton, and Gosselin. The truncated series of this formula are used to approximate functions from the Bernstein space so we establish a bound for the truncation error of this series based on localized sampling without decay assumption. Numerically, we compare our approximation results with the results of Fang and Li's sampling formula. Our formula gives us highly accurate approximations in comparison with the results of Fang and Li's formula. [ABSTRACT FROM AUTHOR]

Abstract: Previous results on certain sampling series have left open if divergence only occurs for certain subsequences or, in fact, in the limit. Here we prove that divergence occurs in the limit. We consider three canonical reconstruction methods for functions in the Paley–Wiener space . For each of these we prove an instance when the reconstruction diverges in the limit. This is a much stronger statement than previous results that provide only divergence. We also address reconstruction for functions in the Hardy space and show that for any subsequence of the natural numbers there exists a function in for which reconstruction diverges in . For two of these sampling series we show that when divergence occurs, the sampling series has strong oscillations so that the maximum and the minimum tend to positive and negative infinity. Our results are of interest in functional analysis because they go beyond the type of result that can be obtained using the Banach–Steinhaus Theorem. We discuss practical implications of this work; in particular the work shows that methods using specially chosen subsequences of reconstructions cannot yield convergence for the Paley–Wiener Space . [Copyright &y& Elsevier]

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 bandwidth (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. [ABSTRACT FROM AUTHOR]

We study the problem of approximating a real-valued functionfby 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. [ABSTRACT FROM PUBLISHER]

In this paper, we analyze sampling approximations of stable linear time-invariant systems for input signals in the Paley-Wiener space \cal PW\pi^{1}. The goal is to approximate a transform Tf by using only the samples of the signal f, which do not necessarily have to be equidistant. We completely characterize the stable linear time-invariant (LTI) systems T and sampling patterns for which the approximation process converges to Tf for all signals in \cal PW\pi^{1}. Furthermore, we prove that for every complete interpolating sequence there exist a stable LTI system and a signal in \cal PW\pi^{1} for which the approximation error increases unboundedly as the number of samples that are used for the approximation is increased. This shows that a more flexible choice of the sampling points instead of equidistant sampling, where such a divergence behavior is known to exist, is not sufficient to overcome the divergence. Thus, sampling based signal processing has fundamental limits. The calculations further indicate that oversampling cannot resolve the convergence problems. [ABSTRACT FROM AUTHOR]

For the Schoenberg (polynomial) B-splines, interesting relations between their functional representation, Dirichlet averages, and difference operators are known. We use these relations to extend B-splines to an arbitrary (infinite) sequence of knots and to higher dimensions. The Fourier domain representation of multidimensional complex B-splines, their two-scale relation and an explicit time domain representation are given for the case of equidistantly distributed knots on rays. In addition, we show that complex B-splines are solutions of the sampling/interpolation problem of complex order z, Re z > 1. [ABSTRACT FROM AUTHOR]

Abstract: In this paper non-equidistant sampling series are studied for bounded bandlimited signals. We consider sampling patterns that are made of the zeros of sine-type functions and analyze the local and global convergence behavior of the sampling series. It is shown that the series converge locally uniformly for bounded bandlimited signals that vanish at infinity. Moreover, we discuss the influence of oversampling on the global approximation behavior and the convergence speed of the sampling series. [Copyright &y& Elsevier]

In this paper, we analyze the approximation of the outputs of linear time-invariant systems by sampling series that use only the samples of the input signal. The samples are disturbed by the threshold operator, which sets all samples with an absolute value smaller than some threshold to zero. We do the analysis for the space of Paley-Wiener signals with absolutely integrable Fourier transform and show for the Hilbert transform that the peak approximation error can grow arbitrarily large for some signals in this space when the threshold approaches zero. This behavior is counterintuitive because one would expect a better behavior if the threshold was decreased. Since we consider oversampling and all kernels from a certain meaningful set, the results are valid not only for one specific approximation process, but for a whole class of approximation processes. Furthermore, we give a game theoretic interpretation of the problem in the setting of a game against nature and show that nature has a universal strategy to win this game. [ABSTRACT FROM AUTHOR]

In practical applications the sole reconstruction of signals by its samples is sometimes not sufficient. Often, some processed version of the signal is of interest and must be approximated by using only the samples of the signal. In this paper, the possible reconstruction kernels are characterized. Then, the convergence behavior of general approximation processes for translation invariant, linear, and bounded operators is analyzed for signals in the Paley-Wiener space and these kernels. It is shown that the Hilbert transform is a universal operator in the sense that the peak value of all possible approximation processes diverges unboundedly for some signal in , regardless of the oversampling factor and the kernel. Furthermore, for all approximation processes and all points in time, there exists an operator such that the approximation process diverges in this point. The results are compared to the approximation behavior of the Hilbert transform, operating on continuous-time signals. Moreover, a simple criterion based on the exponential function as test signal is developed for answering the question of whether or not an approximation process is convergent for a given operator. [ABSTRACT FROM AUTHOR]

Abstract: The convergence of sampling series with non-equidistant sampling points cannot be guaranteed for the Paley–Wiener space if the class of sampling patterns is not restricted. In this paper we consider sampling patterns that are made of the zeros of sine-type functions and analyze the local and global convergence behavior of the sampling series. It is shown that oversampling is necessary for global uniform convergence. If no oversampling is used there exists for every sampling pattern a signal such that the peak value of the approximation error grows arbitrarily large. Furthermore, we use the findings to derive results about the mean-square convergence behavior of the sampling series for bandlimited wide-sense stationary stochastic processes. Finally, a procedure is given to construct functions of sine type and possible sampling patterns. [Copyright &y& Elsevier]

Abstract: In this paper the reconstruction of bandlimited signals by sampling series is analyzed for the case where the samples of the signal are disturbed by the non-linear threshold operator. This operator sets all samples whose absolute value is smaller than some threshold to zero. It is shown that the reconstruction error can grow arbitrarily large, regardless of how small the threshold is chosen, when no oversampling is used. However, with oversampling, it is possible to upper bound the reconstruction error. Additionally, we consider the approximation of outputs of stable linear time-invariant systems, by sampling series that use only the samples that are larger than the threshold, and show that there exist stable linear time-invariant systems for which the approximation error is unbounded, even if oversampling is applied. Finally, we consider the case of non-equidistant sampling. It is possible to draw conclusions about the approximation behavior of the sampling series with threshold operator by analyzing the convergence behavior of the sampling series with permuted sampling points and without threshold operator. Since the latter does not have a good approximation behavior in general, we conjecture that the sampling series with non-equidistant samples and threshold operator has no good approximation behavior either. [Copyright &y& Elsevier]

We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating universality limits for random matrices at the soft edge of the spectrum. [ABSTRACT FROM AUTHOR]

Abstract: In this paper symmetric and non-symmetric Shanon type sampling series that use samples taken at Nyquist rate are analyzed. It is shown that the symmetric series converges uniformly on the whole real axis, i.e., that it is a stable reconstruction process, for the Paley–Wiener space. This result is surprising, because recently it was shown that for a very general class of reconstruction processes a stable reconstruction is not possible for the Paley–Wiener space. However, there are some important differences. The analyzed Shannon type sampling series uses infinitely many samples for signal reconstruction and is not bandlimited. The characteristics stability, bandlimitedness, and the property of perfect reconstruction for a certain subspace are discussed. Furthermore, the corresponding non-symmetric series is analyzed. An explicit example shows that it is not stable. [Copyright &y& Elsevier]

Recently, it was shown with the help of Fourier analysis that by incorporating a Gaussian multiplier into the truncated classical sampling series, one can approximate bandlimited signals of finite energy with an error that decays exponentially as a function of the number of involved samples. Based on complex analysis, we show for a slightly modified operator that this approximation method applies not only to bandlimited signals of finite energy, but also to bandlimited signals of infinite energy, to classes of non-bandlimited signals, to all entire functions of exponential type (including those whose samples increase exponentially), and to functions analytic in a strip and not necessarily bounded. Moreover, the method extends to non-real argument. In each of these cases, the use of 2N + 1 samples results in an error bound of the form Me−αN, where M and α are positive numbers that do not depend on N. The power of the method is illustrated by several examples. [ABSTRACT FROM AUTHOR]

Recently, it was shown with the help of Fourier analysis that by incorporating a Gaussian multiplier into the truncated classical sampling series, one can approximate bandlimited signals of finite energy with an error that decays exponentially as a function of the number of involved samples. Based on complex analysis, we show for a slightly modified operator that this approximation method applies not only to bandlimited signals of finite energy, but also to bandlimited signals of infinite energy, to classes of non-bandlimited signals, to all entire functions of exponential type (including those whose samples increase exponentially), and to functions analytic in a strip and not necessarily bounded. Moreover, the method extends to non-real argument. In each of these cases, the use of 2N +1 samples results in an error bound of the form Me-αN, where M and α are positive numbers that do not depend on N. The power of the method is illustrated by several examples. [ABSTRACT FROM AUTHOR]

Abstract: The Whittaker–Shannon–Kotel''nikov sampling theorem enables one to reconstruct signals f bandlimited to from its sampled values , , in terms of If f is continuous but not bandlimited, one normally considers 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 to f in the -norm for , the classical modulus of continuity being replaced by the averaged modulus of smoothness . The major theorem enables one to sample any bounded signal f belonging to a certain subspace of , the jump discontinuities of which may even form a set of measure zero on . A corollary gives the counterpart of the approximate sampling theorem, now in the -norm. The averaged modulus, so far only studied for functions defined on a compact interval , had first to be extended to functions defined on the whole real axis . Basic tools are the de La Vallée Poussin means and a semi-discrete Hilbert transform. [Copyright &y& Elsevier]

Let $$\mathcal{A}$$ be a symmetric operator with compact resolvent defined in a Hilbert space $$\mathbb{H}.$$ For any fixed $$a \in \mathbb{H}$$ we consider an entire $$\mathbb{H}{\text{ - valued}}$$ function K a which involves the resolvent of $$\mathcal{A}.$$ Associated with K a we obtain, by duality in $$\mathbb{H},$$ a Hilbert space $$\mathcal{H}_{a} $$ of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of $$\mathcal{H}_{a} $$ regardless of the anti-linear mapping which has $$\mathcal{H}_{a} $$ as its range space. There exists also a sampling formula allowing to recover any function in $$\mathcal{H}_{a} $$ from its samples at the sequence of eigenvalues of $$\mathcal{A}.$$ [ABSTRACT FROM AUTHOR]

Let (H,〈·,·〉H) be a complex, separable Hilbert space with orthonormal basis {xn}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 fx(z)=〈K(z),x〉H and let H denote the collection of all such functions fx. 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),xn〉 is analytic for each n∈N and ||K(·)||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. [Copyright &y& Elsevier]

In this paper, we study the aliasing error when a nonbandlimited function is reconstructed by means of prefiltering and sampling. We give the optimal upper bound for the mean aliasing error and show that the error bound can be minimized by a suitable filter. [Copyright &y& Elsevier]

When a sampling theorem holds in wavelet subspaces, sampling expansions can be a good approximation to projection expansions. Even when the sampling theorem does not hold, the scaling function series with the usual coefficients replaced by sampled function values may also be a good approximation to the projection. We refer to such series as hybrid sampling series. For this series, we shall investigate the local convergence and analyze Gibbs phenomenon. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM PUBLISHER]