多元q-Stancu多项式的收敛性及其收敛阶刻画

定义单纯形上的多元q-Stancu多项式,它是著名的Bernstein多项式,q-Bernstein多项式,Stancu多项式的推广.以多元函数的部分连续模及全连续模为度量,建立推广的多元q-Stancu多项式对连续函数的一致收敛定理与收敛阶估计,并以实例加以验证.     

On the rate of pointwise strong summability of orthogonal expansions

There is introduced a modificated local modulus of continuity as a measure of pointwise strong summability of expansion with respect to polynomial-like orthonormal system. The approximation versions and generalizations of the some special cases of the known results of K. Tandori [6], [7] and G. Freud [4] are obtained.     

Sharp estimates in terms of moduli of smoothness for compound cubature rules

Summary In 1980 Dahmen-DeVore-Scherer introduced a modulus of continuity which turns out to reflect invariance properties of compound cubature rules effectively. Accordingly, sharp error bounds are derived, the existence of relevant counterexamples being a consequence of a quantitative resonance principle, established previously.     

q-Baskakov型算子的A-统计逼近

引入一类q-Baskakov型算子,对一个非负正则可求和矩阵A,应用A-统计逼近的理论,研究了这类修正的Korovkin型统计逼近性质.对于0q≤1,借助连续性模,证得这类q-Baskakov型算子的收敛速度要优于q-Baskakov算子.     

QUASI-MODULUS OF CONTINUITY AND BEST APPROXIMATIONS

In this paper a concept of the quasi-modulus of continuity of the functions of L^2(T) on the unit circle T is introduced, and the relations between it and modulus of continuity are discussed. And we give a sufficient condition such that the best uniform approxi marion of the continuous functions on T is continuous.     

Moduli of continuity and rearrangements

It is proved that the modulus of continuity of a function in L_p(0, 1) majorizes the modulus of continuity of its decreasing rearrangement.     

Dunkl generalization of Szász beta‐type operators

The goal in the paper is to advertise Dunkl extension of Szász beta‐type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second‐order modulus of continuity, the Lipschitz class functions, Peetre's K‐functional, and modulus of weighted continuity by Dunkl generalization of Szász beta‐type operators.     

多元Stancu多项式与连续模

本文研究单纯形上多元Stancu多项式与连续模之间的关系,证明了Stancu多项式具有保持连续模的性质,推广了一元Bernstein多项式的相应结果.同时,利用多元函数的Ditzian-Totik连续模估计Stancu多项式逼近多元连续函数速度的上界和下界,得到一个使得逼近速度为O(n-a)(0相似文献   

Generalized Logan’s Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson’s Inequality in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Superscript>3</Superscript>)

We study Jackson's inequality between the best approximation of a function f ∈ L₂(R³) by entire functions of exponential spherical type and its generalized modulus of continuity. We prove Jackson's inequality with the exact constant and the optimal argument in the modulus of continuity. In particular, Jackson's inequality with the optimal parameters is obtained for classical modulus of continuity of order r and Thue-Morse modulus of continuity of order r ∈ N. These results are based on the solution of the generalized Logan problem for entire functions of exponential type. For it we construct a new quadrature formulas for entire functions of exponential type.     

An estimate for the modulus of continuity of a quaternion singular Cauchy integral

We establish an upper bound for the modulus of continuity of a quaternion singular Cauchy integral in terms of the modulus of continuity of the integrand and a metric characteristic of a curve.     

Numerical solution of stochastic differential equations by second order Runge–Kutta methods

In this paper we propose the numerical solutions of stochastic initial value problems via random Runge–Kutta methods of the second order and mean square convergence of these methods is proved. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced. Expectation and variance of the approximating process are computed. Numerical examples show that the approximate solutions have a good degree of accuracy.     

Characterization of holomorphic functions in terms of their moduli

It was shown in (Boche, H. and Pohl, V., 2005, Spectral factorization in the disk algebra. Complex Variables. Theory and Applications, 50, 383–387.) that if the modulus |f| of a function is continuous in the closure of the unit disk, the function f itself needs not to be continuous there, in general. This article shows that if the modulus of continuity of a function is a weak regular majorant, the continuity of the modulus always implies the continuity of the function itself.     

Degree of Approximation for Bivariate Chlodowsky–Szasz–Charlier Type Operators

We consider a combination of Chlodowsky polynomials with generalized Szasz operators involving Charlier polynomials. We give the degree of approximation for these bivariate operators by means of the complete and partial modulus of continuity, and also by using weighted modulus of continuity. Furthermore, we construct a GBS (Generalized Boolean Sum) operator of bivariate Chlodowsky–Szasz–Charlier type and estimate the order of approximation in terms of mixed modulus of continuity.     

On the modulus of continuity of the solution to the Dirichlet problem at a regular boundary point

A linear elliptic equation of second order with coefficients satisfying a Dini condition is considered in the paper. The modulus of continuity of a solution at a regular boundary point is investigated. An estimate for the modulus of continuity in terms of the Wiener capacity is obtained.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 67–72, July, 1972.     

SOME FUNCTIONAL LIMIT THEOREMS FOR THE INFINITE SERIES OF OU PROCESSES

This paper obtains functional modulus of continuity and Strassen's functional LIL of the infinite series of independent Ornstein-Uhlenbeck processes, which also imply the Levy's exact modulus of continuity and LIL of this process respectively.     

Numerical solution of random differential equations: A mean square approach

This paper deals with the construction of numerical solutions of random initial value differential problems by means of a random Euler difference scheme whose mean square convergence is proved based on conditions expressed in terms of the mean square behavior of the right-hand side of the underlying random differential equation. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced and illustrative examples and possibilities are included. Expectation and variance of the approximating process are computed.     

<Emphasis Type="Italic">q</Emphasis>- Lupas Kantorovich operators based on Polya distribution

The purpose of the present paper is to introduce a Kantorovich modification of the q-analogue of the Stancu operators defined by Nowak (J Math Anal Appl 350:50–55, 2009). We study a local and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Further A-statistical convergence properties of these operators are investigated. Next, a bivariate generalization of these operators is introduced and its rate of convergence is discussed with the aid of the partial and complete modulus of continuity and the Peetre‘s K-functional.     

Некоторые точные нер авенства в теории при ближения функций

Exact inequalities are obtained that illuminate the interrelation between best polynomial approximations of functions, analytic in the disk and the modulus of continuity of the derivatives of the boundary values of these functions.For various classes of functions exact estimates are given for the derivative of a function by means of the modulus of continuity of this function and the modulus of continuity of its second derivative.As application, exact inequalities are deduced, analogous to the well-known Bernstein and Hardy inequalities.     

Fourier coefficients of functions with a given modulus of continuity

In this paper, we study the role of the convexity condition for the modulus of continuity in the problem of finding an upper bound for the Fourier coefficients taken over the class of functions with a given modulus of continuity. Also, we solve the problem of the Fourier coefficients for the Rademacher system.     

Modulus of continuity of the solution to the Dirichlet problem for a second-order parabolic equation

The modulus of continuity of the solution to the Dirichlet problem is investigated for a second-order parabolic equation at a regular boundary point. A bound for the modulus of continuity is obtained in terms of the capacity. The coefficients of the equation are required to satisfy a Dini condition (uniformly).Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 587–593, April, 1976.