Summation of Some Trigonometric and Schlömilch Series

In this paper we consider trigonometric series in terms of the Riemann zeta function and related functions of reciprocal powers. The obtained closed form formulas we apply to the evaluation of the Riemann zeta function and related functions of reciprocal powers. One can establish recursive relations for them and relations between any two of those functions. These closed formulas enable us also to find sums of some Schlömilch series. We give an example which shows how the convergence of a trigonometric series can be accelerated by applying Krylov's method and our formula (7).     

A Modification of Hermite Sampling with a Gaussian Multiplier

The Hermite sampling series is used to approximate bandlimited functions. In this article, we introduce two modifications of Hermite sampling with a Gaussian multiplier to approximate bandlimited and non-bandlimited functions. The convergence rate of those modifications is much higher than the convergence rate of Hermite sampling. Based on complex analysis, we establish some error bounds for approximating different classes of functions by these modifications. Theoretically and numerically, we demonstrate that the approximation by these modifications is highly efficient.     

Frame Analysis of Irregular Periodic Sampling of Signals and Their Derivatives

Given a bandlimited signal, we consider the sampling of the signal and some of its derivatives in a periodic manner. The mathematical concept of frames is utilized in the analysis of the properties of the sequence of sampling functions. The frame operator of this sequence is expressed as a matrix-valued function multiplying a vector-valued function. An important property of this matrix is that the maximum and minimum eigenvalues are equal (in some sense) to the upper and lower frame bounds. We present a method for finding the dual frame and, thereby, a method for reconstructing the signal from its samples. Using the matrix approach we prove that the sequence of sampling functions is always complete in the cases of critical sampling and oversampling. A sufficient condition for the sequence of sampling functions to constitute a frame is derived. We show that if no sampling of the signal itself is involved, the sampling is not stable and cannot be stabilized by oversampling. Examples are considered, and the frame bounds in the case of sampling of the signal and its first derivative are calculated explicitly. Finally, the matrix approach can be similarly applied to other problems of signal representation.     

Stability analysis for several second-order Sigma—Delta methods of coarse quantization of bandlimited functions

We investigate the stability and robustness properties of a family of algorithms used to “coarsely quantize” bandlimited functions. The algorithms we will consider are one-bit second-orderΣΔA-quantization schemes and some modified versions of these. We prove that there exists a bounded region that remains positively invariant under the two-dimensional piecewise-affine discrete dynamical system associated with each of these quantizers. Moreover, this bounded region can be constructed so that it is robust under small changes in the quantizer. We also show some interesting properties of the resulting binary sequences.     

A new class of polynomials associated with Bernstein and beta polynomials

The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd.     

一类有理插值曲面模型及其可视化约束控制

本文构造一类新的基于函数值和偏导数值的双变量加权混合有理插值样条.与已有的有理插值样条相比,这类新的有理插值样条具有以下四方面的特性,其一,插值函数可以由简单的对称基函数来表示;其二,对任何正参数,插值函数满足C1连续,而且,在不限制参数取值的条件之下,插值曲面保持光滑;其三,插值函数不但含有参数,而且带有加权系数,增加了插值函数的自由度;其四,插值曲面的形状随着参数与加权系数的变化而变化.同时,本文讨论此类插值曲面的性质,包括基函数的性质、积分加权系数的性质和插值函数的边界性质.此类插值函数的优势在于,不改变给定插值数据的前提下,通过选择合适的参数和不同的加权系数,对插值区域内的任意点的函数值进行修改.因此可将其应用于曲面设计,根据实际设计需要,自由地修改曲面形状.数值实验表明,此类新的有理样条插值具有良好的约束控制性质.     

Truncation,amplitude, and jitter errors on R for sampling series derivatives

In this paper, we investigate the error analysis of the derivative of the classical sampling theorem of bandlimited functions. We consider truncation, amplitude, and time-jitter errors. Both pointwise and uniform estimates are given. We derive analogues of the results of Piper (1975), Brown (1969), Jagerman (1966) and Li (1998) in a generalized manner. The amplitude and time-jitter errors are studied in the view of the works of Butzer (1983) and Butzer et al. (1988), provided that the bandlimited function satisfies some decay properties.     

Paley-Wiener subspace of vectors in a Hilbert space with applications to integral transforms

The goal of this article is to introduce an analogue of the Paley-Wiener space of bandlimited functions, PWω, in Hilbert spaces and then apply the general result to more specific examples. Guided by the role that the differentiation operator plays in some of the characterizations of the Paley-Wiener space, we construct a space of vectors using a self-adjoint operator D in a Hilbert space H, and denote this space by PWω(D). The article can be virtually divided into two parts. In the first part we show that the space PWω(D) has similar properties to those of the space PWω, including an analogue of the Bernstein inequality and the Riesz interpolation formula. We also develop a new characterization of the abstract Paley-Wiener space in terms of solutions of Cauchy problems associated with abstract Schrödinger equations. Finally, we prove two sampling theorems for vectors in PWω(D), one of which uses the notion of Hilbert frames and the other is based on the notion of variational splines in H. In the second part of the paper we apply our abstract results to integral transforms associated with singular Sturm-Liouville problems. In particular we obtain two new sampling formulas related to one-dimensional Schrödinger operators with bounded potential.     

Geometry of interpolation sets in derivative free optimization

We consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated. We show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods. These bounds involve a constant that reflects the quality of the interpolation set. The main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small, and at least uniformly bounded. This constant is often described through the basis of Lagrange polynomials associated with the interpolation set. We provide an alternative, more intuitive, definition for this concept and show how this constant is related to the condition number of a certain matrix. This relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded. We also derive bounds on the error between the model and the function and between their derivatives, directly in terms of this condition number and of this geometry constant.     

Interpolation by neural network operators activated by ramp functions

In this paper, a family of interpolation neural network operators are introduced. Here, ramp functions as well as sigmoidal functions generated by central B-splines are considered as activation functions. The interpolation properties of these operators are proved, together with a uniform approximation theorem with order, for continuous functions defined on bounded intervals. The relations with the theory of neural networks and with the theory of the generalized sampling operators are discussed.     

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.     

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.     

Reconstruction of Bandlimited Functions from?Unsigned Samples

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its Nyquist rate, and may not necessarily be possible if the samples are taken at less than twice the Nyquist rate. In the case of uniform samples, we also describe an FFT-based algorithm to perform the reconstruction. We prove that it converges exponentially rapidly in the number of samples used and examine its numerical behavior on some test cases.     

Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree Δ. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for this problem, the Wang–Swendsen–Kotecký (WSK) algorithm. The second author recently proved that the WSK algorithm quickly converges to the desired distribution when k11Δ/6. We study how far these positive results can be extended in general. In this note we prove the first non-trivial results on when the WSK algorithm takes exponentially long to reach the stationary distribution and is thus called torpidly mixing. In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3k<Δ/(20logΔ), and on a family of planar graphs for any number of colors. We also give a family of graphs for which, despite their small chromatic number, the WSK algorithm is not ergodic when kΔ/2, provided k is larger than some absolute constant k₀.     

Double sampling derivatives and truncation error estimates

This paper investigates double sampling series derivatives for bivariate functions defined on R² that are in the Bernstein space. For this sampling series, we estimate some of the pointwise and uniform bounds when the function satisfies some decay conditions. The truncated series of this formula allow us to approximate any order of partial derivatives for function from Bernstein space using only a finite number of samples from the function itself. This sampling formula will be useful in the approximation theory and its applications, especially after having the truncation error well-established. Examples with tables and figures are given at the end of the paper to illustrate the advantages of this formula.     

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.     

Vector-valued fractal functions: Fractal dimension and fractional calculus

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.     

Characterization of local sampling sequences for spline subspaces

The sampling theorem is one of the most powerful tools in signal analysis. It says that to recover a function in certain function spaces, it suffices to know the values of the function on a sequence of points. Most of known results, e.g., regular and irregular sampling theorems for band-limited functions, concern global sampling. That is, to recover a function at a point or on an interval, we have to know all the samples which are usually infinitely many. On the other hand, local sampling, which invokes only finite samples to reconstruct a function on a bounded interval, is practically useful since we need only to consider a function on a bounded interval in many cases and computers can process only finite samples. In this paper, we give a characterization of local sampling sequences for spline subspaces, which is equivalent to the celebrated Schönberg-Whitney Theorem and is easy to verify. As applications, we give several local sampling theorems on spline subspaces, which generalize and improve some known results.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

Bounded rank-one perturbations in sampling theory

Sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points. In this paper we derive an orthogonal sampling theory and associated Lagrange interpolation formulae from a family of bounded rank-one perturbations of a self-adjoint operator that has only discrete spectrum of multiplicity one.