Bounded mean oscillation and bandlimited interpolation in the presence of noise

We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties.     

On Generalized Gaussian Quadratures for Exponentials and Their Applications

We introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions.We use a generalization of a representation theorem due to Carathéodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. For a given accuracy ε, selecting an eigenvalue close to ε yields an approximate quadrature with that accuracy. To compute its weights and nodes, we present a new fast algorithm.These new quadratures can be used to approximate and integrate bandlimited functions, such as prolate spheroidal wave functions, and essentially bandlimited functions, such as Bessel functions. We also develop, for a given precision, an interpolating basis for bandlimited functions on an interval.     

Approximation of signals from local averages

This work is concerned with approximation of a signal from local averages. It improves a result of Butzer and Lei [P.L. Butzer, J. Lei, Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal. 4 (2000) 245–255].     

Approximation of bandlimited functions

Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae.     

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory.     

Landau's necessary density conditions for LCA groups

We derive necessary conditions for sampling and interpolation of bandlimited functions on a locally compact abelian group in line with the classical results of H. Landau for bandlimited functions on Rd. Our conditions are phrased as comparison principles involving a certain canonical lattice.     

A Survey on Recent Results of the Mathematical Seminar in Perugia, inspired by the Work of Professor P.L. Butzer

Here we give a survey on recent results in approximation theory obtained by a group of mathematicians of the University of Perugia, inspired by the work of Professor Butzer. We collect some results concerning linear integral operators in various settings.     

Statistical approximation properties of the Durrmeyer type q‐Bleimann,Butzer, and Hahn operators

In this paper, we introduce a Durrmeyer‐type generalization of q‐Bleimann, Butzer, and Hahn operators based on q‐integers and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. We also compute rates of statistical convergence of these q‐type operators by means of the modulus of continuity and Lipschitz‐type maximal function, respectively. Copyright © 2011 John Wiley & Sons, Ltd.     

A Best Asymptotic Constant Related to the Operators of Bleimann, Butzer and Hahn

We determine the best asymptotic constant in an estimation of the second central moment of the Bleimann, Butzer and Hahn operators L _n yielding some new results concerning their rate of convergence. In particular, we find $$\lim_{n\in\mathbb{N}}\sup_{0 < x < \infty}\frac{n\,L_{n}((\cdot-x)^{2} ;x)}{x(1+x)^{2}}$$ .     

Half Sampling on Bipartite Graphs

On a bipartite graph G we consider the half sampling problem of uniquely recovering a function from its values on the even vertices, under the appropriate half bandlimited assumption with respect to a Laplacian on the graph. We discuss both finite and infinite graphs, give the appropriate definition of “half bandlimited” that involves splitting the mid frequency, and give an explicit solution to the problem. We discuss in detail the example of a regular tree. We also consider a related sampling problem on graphs that are generated by edge substitution.     

Sampling of Operators

Sampling and reconstruction of functions is a fundamental tool in science. We develop an analogous sampling theory for operators whose Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for in this sense bandlimited operators and show that our results generalize both, the classical sampling theorem, and the fact that a time-invariant operator is fully determined by its impulse response.     

Korovkin type theorem for non‐tensor Balázs type Bleimann,Butzer and Hahn operators

In this paper, we prove a certain Korovkin type approximation theorem by introducing new test functions. We introduce the non‐tensor Balázs type Bleimann, Butzer and Hahn operators and give the approximation property by using this new Korovkin theorem. Furthermore, we obtain the rate of convergence of these operators by means of modulus of continuity. Finally, we state the multivariate version of the abovementioned Korovkin type theorem. Copyright © 2014 John Wiley & Sons, Ltd.     

Limiting Case for Interpolation Spaces Generated by Holomorphic Semigroups

tA )_t≥0 on X. We denote by X_A(θ,p), for 0 < θ < 1, the interpolation spaces of X and D(A) introduced by Berens and Butzer in [1]. These spaces play an important role in the approximation theory ( [2] ) as well as in the study of the abstract parabolic equation u′ (t) = Au(t) + f(t) ( [3] ). It has been proved in [6] that these spaces are meaningful (and still enjoy relevant properties) also in the case θ = 0. In this paper we continue the study of [6] and prove new interesting properties of these spaces.     

On the analytic extension of Stirling numbers of the first kind

We present an analytic extension of the unsigned Stirling numbers of the first kind that is in a certain sense unique in its coincidence with the Stirling polynomials. We examine and compare our extension to previous extensions of (signed) Stirling numbers of the first kind given by Butzer et al. (2007, J. Difference Equ. Appl., 13) and of the unsigned numbers given by Adamchik (1997, J. Comput. Appl. Math., 79). We also see a connection to the Riemann zeta function.     

Local saturation of conservative operators

We state a result about the local saturation of sequences of linear operators that preserve the sign of the k-th derivative of the functions. We apply it to the well known approximation operators of Bernstein, Szász–Mirakjan, Meyer–König and Zeller, and Bleimann, Butzer and Hahn.     

Concentration Problems for Bandpass Filters in Communication Theory over Disjoint Frequency Intervals and Numerical Solutions

The concentration problem of maximizing signal strength of bandlimited and timelimited nature is important in communication theory. In this paper we consider two types of concentration problems for the signals which are bandlimited in disjoint frequency-intervals, which constitute a band-pass filter. For the first type the problem is to determine which members of L ²(−∞,∞) lose the smallest fraction of their energy when first timelimited and then bandlimited. For the second type the problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted to a given time interval. For both types of problems, basic theoretical properties and numerical algorithms for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L ²(−∞,∞) are also proved. Numerical computations are carried out which corroborate the theory. Relationship between eigenvalues of these two types of problems is also established. Several properties of eigenvalues of both types of problems are proved.     

A Unified Approach to Generalized Stirling Functions

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, k-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [16]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed,which extend the corresponding results about the Stirling numbers shown in [21] to the defined Stirling functions.     

ON TERM BY TERM DYADIC DIFFERENTIABILITY OF WALSH—KACZMARZ SERIES

The aim of this paper is to prove the following theorem concerning the term by term differentiation the-orem of Walsh-Kaczmarz series. Let (ck) be a decreasing real sequence withare integrable functions and f(x) is a. e. dyadic (or Butzer and Wagner) differentiate withThe function Kk means the kth Walsh-Kaczmarz function.     

The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics

In this paper, we establish a Mellin version of the classical Parseval formula of Fourier analysis in the case of Mellin bandlimited functions, and its equivalence with the exponential sampling formula (ESF) of signal analysis, in which the samples are not equally spaced apart as in the classical Shannon theorem, but exponentially spaced. Two quite different examples are given illustrating the truncation error in the ESF. We employ Mellin transform methods for square-integrable functions.