Some results on zeros and uniqueness of difference-differential polynomials

We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions.We also investigate the uniqueness problems of difference...     

Multi-node higher order expansions of a function

By using the values and higher derivatives of a function at the given nodes, a kind of multi-node higher order expansion of the function is presented. The error terms of the expansions are given. Particular examples are the extensions of the Taylor polynomials, Bernstein polynomials and Lagrange interpolation polynomials. The expansions are numerical approximation polynomials and very useful particular for the functions for which the higher derivatives can be obtained easily.     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

Derivative polynomials and closed-form higher derivative formulae

In a recent paper, Adamchik [1] expressed in a closed-form symbolic derivatives of four functions belonging to the class of functions whose derivatives are polynomials in terms of the same functions. In this sequel, simple closed-form higher derivative formulae which involve the Carlitz-Scoville higher order tangent and secant numbers are derived for eight trigonometric and hyperbolic functions.     

Approximation properties of the Vallée-Poussin means of partial sums of a mixed series of Legendre polynomials

We estimate the order of weighted approximations of functions and their derivatives by using the means of mixed series of Legendre polynomials. As the main result, we obtain estimates of the order of approximation of a function and its derivatives by the Vallé-Poussin means and their derivatives.     

Approximation by Bernstein polynomials at the discontinuity points of derivatives

We obtain asymptotic formulas for the deviation of Bernstein polynomials from functions at the first-kind discontinuity points of the highest derivatives of odd order.     

Ba空间中导数型的Jackson定理

本文研究了Ｂａ空间中的函数及其各阶导函数用多项式同时逼近的问题,证明了导数型的Ｊａｃｋｓｏｎ定理。     

Hobson's formula for Dunkl operators and its applications

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials.     

Solution of the three‐dimension wave equation by using wave polynomials

The paper demonstrates a specific power series expansion technique to solve the three‐dimensional wave equation. As solving functions, so‐called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry without strength restrictions. The wave polynomials are defined. Recurrent formulas for the wave polynomials and their derivatives are obtained. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of Szász-Beta operators

In the present article we find the hypergeometric representation of the Szász-Beta operators and obtain the moments using confluent hypergeometric functions, which can be related to generalized Laguerre polynomials. Then we estimate a convergence result for functions having bounded derivatives.     

Several formulas for special values of the Bell polynomials of the second kind and applications

In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.     

“Restricted derivatives” approximation to functions with derivatives outside the range

The problem of best uniform approximation by polynomials with restricted ranges of some of their derivatives to functions not satisfying the same restrictions is treated. Results concerning the number of alternations of the best approximating polynomial are derived, and the impossibility of approximating arbitrarily closely functions with one derivative outside the range is proved.     

Some algorithms for numerical quadrature using the derivatives of the integrand in the integration interval

Some quadrature formulae using the derivatives of the integrand are discussed. As special cases are obtained generalizations of both the ordinary and the modified Romberg algorithms. In all cases the error terms are expressed in terms of Bernoulli polynomials and functions.     

关于用Hermitfe插值逼近函数及其导数的点态估计

本文给出了以第一类Chebyshev多项式的零点为节点的Hermitfe插值多项式来逼近函数及其导数时的逼近阶的点态估计式,该结果回答了谢庭藩教授在文[4]中提出的问提.     

On the pointwise estimations of approximation of functions and their derivatives by Hermite interpolation

The object of this paper is to establish the pointwise estimations of approximation of functions in C¹ and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in general.     

On Local Smoothness Classes of Periodic Functions

We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. We construct a sequence of operators whose values are (global) trigonometric polynomials, and yet their behavior at different points reflects the behavior of the target function near each of these points. In addition to being localized, our operators preserve trigonometric polynomials of degree commensurate with the degree of polynomials given by the operators. Our constructions are “universal;” i.e., they do not require an a priori knowledge about the smoothness of the target functions. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of derivatives of different order.     

The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients

This paper deals with the numerical solution of classes of fractional convection–diffusion equations with variable coefficients. The fractional derivatives are described based on the Caputo sense. Our approach is based on the collocation techniques. The method consists of reducing the problem to the solution of linear algebraic equations by expanding the required approximate solution as the elements of shifted Legendre polynomials in time and the Sinc functions in space with unknown coefficients. The properties of Sinc functions and shifted Legendre polynomials are then utilized to evaluate the unknown coefficients. Several examples are given and the numerical results are shown to demonstrate the efficiency of the newly proposed method.     

亚纯函数系数微分方程的解和小函数的关系

研究了一类亚纯函数为系数的二阶非齐次线性微分方程的解及其微分多项式和小函数的关系,并得到了这类微分方程解以及解的一阶,二阶导数与微分多项式的不动点性质.     

Chromatic series for functions of slow growth

The theory of chromatic derivatives leads to chromatic series which replace Taylor's series for bandlimited functions. For such functions, these series have a global convergence property not shared by Taylor's series. In this work the theory is extended to bandlimited functions of slow growth. This includes many signals of practical importance such as polynomials, periodic functions and almost periodic functions. This extension also enables us to get improved local convergence results for chromatic series.     

О восстановлении пер иодических функций н ескольких переменных по значен иям в узлах теоретико числовых сеток

For multivariate functions with bounded mixed derivatives, construction problems of approximating polynomials are treated making use of the values at the nets with fixed cardinality. Problems of construction of good quadrature formulae for the corresponding functional classes are treated, as well.