Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Zero Angles

Let C be the extended complex plane; G C a finite Jordan with 0 G; w= (z) the conformal mapping of G onto the disk normalized by . Let us set , and let be the generalized Bieberbach polynomial of degree n for the pair (G,0), which minimizes the integral in the class of all polynomials of degree not exceeding n with . In this paper we study the uniform convergence of the generalized Bieberbach polynomials with interior and exterior zero angles and determine its dependence on the properties of boundary arcs and the degree of their tangency.     

On Rates of Uniform Convergence of Lower Extreme Generalized Order Statistics

Under a von Mises-type condition the joint distribution of suitable normalized lower extreme generalized order statistics converges w.r.t. the variational distance to the asymptotic joint distribution of lower extreme order statistics. Rates of uniform convergence are established. It turns out that the rates of uniform convergence known for ordinary extremes carry over to lower generalized extremes. Finally, models of Weibull type are concerned, where uniform rates are used in connection with model approximations in order to simplify statistical inference.AMS 2000 Subject Classification. Primary—60G70     

On Uniform Convergence Structures

In this paper we study a general notion of a uniform convergence structure. Morphisms of various supremum-complete subsets of the complete lattice of all uniform convergence structures are investigated. They provide a satisfactory framework for the uniformization of arbitrary convergences.     

On the Convergence Regions of Generalized Accelerated Overrelaxation Method for Linear Complementarity Problems

In this paper, we use a generalized Accelerated Overrelaxation (GAOR) method and analyze the convergence of this method for solving linear complementarity problems. Furthermore, we improve on the convergence region of this method with acknowledgement of the maximum norm. A numerical example is also given, to illustrate the efficiency of our results.     

Convergence of Bieberbach polynomials in domains with interior cusps

We extend the results on the uniform convergence of Bieberbach polynomials for domains with certain interior zero angles (outward pointing cusps) and show that they play a special role in the problem. Namely, we construct a Keldysh-type example on the divergence of Bieberbach polynomials at an outward pointing cusp and discuss thecritical order of tangency at this interior zero angle, separating the convergent behavior of Bieberbach polynomials from the divergent one for sufficiently thin cusps. Research of both authors was supported in part by the National Science Foundation grant DMS-9707359. Research of the second author was also supported in part by the National Science Foundation grant DMS-9970659.     

关于Bernstein多项式的绝对收敛性

该文研究Ｂｅｒｎｓｔｅｉｎ多项式的绝对收敛性．证明了，对每个ｘ∈［０，１］，一个有界变差函数的Ｂｅｒｎｓｔｅｉｎ多项式序列是绝对｜Ｃ，１｜可和的，而且给出了Ｂｅｒｓｔｅｉｎ多项式序列的绝对｜Ｃ，１｜和式的余项的估计．     

On the Uniform Convergence of Walsh-Fourier Series

We study the uniform convergence of Walsh-Fourier series of functions on the generalized Wiener class BV (p(n)↑∞) This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Convergence of Bernstein-Stancu Polynomials in the Variation Seminorm

This article studies the variation detracting property and rate of approximation of the Bernstein-Stancu polynomials in the space of functions of bounded variation with respect to the variation seminorm. Moreover, we will present Voronovskaya-type theorems for Bernstein-Stancu polynomials B_{n, α, β}f and for the first derivative of these polynomials. Finally we include some graphical examples.     

Best Uniform Approximation of Complex-Valued Functions by Generalized Polynomials Having Restricted Ranges

We study the uniform best restricted ranges approximations of complex-valued functions by generalized polynomials. The theory, generalizing the real-valued case, embraces the theorems of existence, characterization, uniqueness, and strong uniqueness.     

On Uniform Boundedness and Uniform Asymptotics for Orthogonal Polynomials on the Unit Circle

The known conditions due to G. Baxter, Ya. L. Geronimus, and B. L. Golinskii which guarantee the uniform boundedness and/or uniform asymptotic representation for orthonormal polynomials on the unit circle are under consideration. We show that these conditions are in general not necessary. We discuss the relation between the orthonormal polynomials on the unit circle, the best approximations, and absolutely convergent Fourier series.     

复域中高阶Fejér插值的一致收敛性

对于复域中满足某种条件的Jordan区域D和函数f∈B(D),证明了基于Fejer点的高阶Fejer插值多项式一致收敛于对应的函数f(z)于D上.本文中的这些定理推广了某些已知的结果.     

On Sobolev Orthogonality for the Generalized Laguerre Polynomials

The orthogonality of the generalized Laguerre polynomials, {L^(α)_n(x)}_n0, is a well known fact when the parameterαis a real number but not a negative integer. In fact, for −1<α, they are orthogonal on the interval [0, +∞) with respect to the weight functionρ(x)=x^αe^−x, and forα<−1, but not an integer, they are orthogonal with respect to a non-positive definite linear functional. In this work we will show that, for every value of the real parameterα, the generalized Laguerre polynomials are orthogonal with respect to a non-diagonal Sobolev inner product, that is, an inner product involving derivatives.     

广义Dirichlet级数的收敛性

本文把决定Dirichlet级数收敛横坐标的Kojima—Knopp公式推广到复指数Dirichlet级数情形.     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].