RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP --In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

Hausdorff operators on Euclidean spaces

Hausdorff operator is an important operator raised from the dilation on Euclidean space and rooted in the classical summability of number series and Fourier series. It is also connected to many well known operators in real and complex analysis. This article is a survey of some recent developments and extensions on the Hausdorff operator. Particularly, various boundedness properties of the Hausdorff operators, studied recently by our research group, are addressed.     

A Longitudinal Study of the Effects of Family Background Factors on Mathematics Achievements Using Quantile Regression

Quantile regression is gradually emerging as a powerful tool for estimating models of conditional quantile functions, and therefore research in this area has vastly increased in the past two decades. This paper, with the quantile regression technique, is the first comprehensive longitudinal study on mathematics participation data collected in Alberta, Canada. The major advantage of longitudinal study is its capability to separate the so-called cohort and age effects in the context of population studies. One aim of this paper is to study whether the family background factors alter performance on the mathematical achievement of the strongest students in the same way as that of weaker students based on the large longitudinal sample of 2000, 2001 and 2002 mathematics participation longitudinal data set. The interesting findings suggest that there may be differential family background factor effects at different points in the mathematical achievement conditional distribution.     

Factorizations of Finite Groups and Related Topics

This is a survey on the recent progress in the theory of finite groups with factorizations and around it,done by the author and his coauthors,and this has no pretensions to cover all topics in this wide area of research.In particular,we only touch the great consequences of the fundamental paper of Liebeck,Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with factorizations.In each case the reader can find additional references at the end of Section 1.Some of the methods of investigation can be used to obtain information about finite groups in general,nilpotent algebras and related nearrings.     

Theorems of Erdős-Ko-Rado type in geometrical settings

The original Erds-Ko-Rado problem has inspired much research. It started as a study on sets of pairwise intersecting k-subsets in an n-set, then it gave rise to research on sets of pairwise non-trivially intersecting k-dimensional vector spaces in the vector space V (n, q) of dimension n over the finite field of order q, and then research on sets of pairwise non-trivially intersecting generators and planes in finite classical polar spaces. We summarize the main results on the Erds-Ko-Rado problem in these three settings, mention the Erds-Ko-Rado problem in other related settings, and mention open problems for future research.     

Vol.4 Ser.A No.6 1983——CONTENTS AND ABSTRACTS

Multipliers for|C,1|Summability of Fourier SeriesJi Yaoming(嵇耀明)(Hangzhou University)The main results of this note improve the conditions of all previously known theorems on the|C,1|summability factors of Fourier series.     

THE BOUNDEDNESS OF CERTAIN OPERATORS ON HOLDER AND SOBOLEV SPACES

The main topic in this note is to discuss the boundedness of pseudo-differential operators and para-product nptfators on ihe Holder and Sobotev space;, respectively It is the preparation of the thoery of Gibbs - Butzer deffereftital operators and differential equations.     

Function characterizations via commutators of Hardy operator

This paper is a summary of the research on the characterizations of central function spaces by the author and his collaborators in the past ten years.More precisely,the author gives some characterizations of central Campanato spaces via the boundedness and compactness of commutators of Hardy operator.     

A Brief Introduction to Fibrewise Topological Spaces Theory

Fibrewise topological spaces theory,presented in the recent 20 years,is a new branch of mathematics developed on the basis of General Topology,Algebra topology and Fibrewise spaces theory.It is associated with differential geometry,Lie groups and dynamical systems theory.From the perspective of Category theory,it is in the higher category of general topological space,so the discussion of new properties and characteristics of the variety of fibre topological space has more important significance.This paper introduces the process of the origin and development of Fibrewise topological spaces theory.Then,we study the main contents and important results in this branch.Finally,we review the research status of Fibrewise topological spaces theory and some important topics.     

INTERPOLATION OF Ba SPACES

In this paper we advance the interpolation theorems of linear operator in Ba spaces. The conception of Ba spaces was first put forward by Ding Xiaxi in his discussion on the prior estimate of Laplace operator in some classical domains and in his study of the embedding theorem of Orlicz-Sobolev spaces, higher dimensional singular     

Jackson-Type Inequality for Doubling Weights on the Sphere

In the one-dimensional case, Jackson's inequality and its converse for weighted algebraic polynomial approximation, as well as many important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumption on the weights. In most cases this minimal assumption is the doubling condition. In this paper, we establish Jackson's theorem and its Stechkin-type converse for spherical polynomial approximation with respect to doubling weights on the unit sphere.     

Approximation properties of a multilayered feedforward artificial neural network

We prove that an artificial neural network with multiple hidden layers and akth-order sigmoidal response function can be used to approximate any continuous function on any compact subset of a Euclidean space so as to achieve the Jackson rate of approximation. Moreover, if the function to be approximated has an analytic extension, then a nearly geometric rate of approximation can be achieved. We also discuss the problem of approximation of a compact subset of a Euclidean space with such networks with a classical sigmoidal response function.Dedicated to Dr. C.A. Micchelli on the occasion of his fiftieth birthday, December 1992Research supported in part by AFOSR Grant No. 226 113 and by the AvH Foundation.     

Local smoothing of uniformly smooth maps

We solve the problem on the uniform approximation of uniformly continuous (smooth) maps by maps having the maximum possible local and uniform smoothness. In particular, we prove that each uniformly continuous map of the Hilbert space l ₂ into itself can be approximated by locally infinitely differentiable maps having a Lipschitz derivative.     

Cultivating a research imperative: Mentoring mathematics at Stockholms Högskola, 1882–1887

Though the central role of Gösta Mittag-Leffler in the promotion of specialized, research-oriented mathematics at Stockholms Högskola is widely acknowledged, the specific social and technical means by which he sought to cultivate a fledgling research community there during the early- to mid-1880s have received little attention. In particular, a detailed study of the relationship of his own research activity to that of his first Swedish students is absent from the existing literature.Through the juxtaposition of their research activities and unpublished correspondence, this paper explores Mittag-Leffler's active and deliberate efforts to engage his students Ivar Bendixson and Edvard Phragmén in open problems within his own research agenda, support them through his institutional connections, and instill within them norms concerning research ideologies, practices of communication and criticism, and frameworks for shared knowledge. It also illuminates the extent to which his teachings took root in at least one student to emerge from his program, who would perpetuate the mathematical practices set in place by his teacher and set forth on the international stage to promote his newly-acquired system of values.     

Polynomial approximation of functions with given order of thekth generalized modulus of smoothness

The problem of approximation by algebraic polynomials is considered on function classes characterized by the value of thekth generalized modulus of smoothness defined by the Jacobi generalized shift operator. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 425–436, March, 1998.     

Bernstein-Kantorovich算子导数的逼近

给出了Bernstein-Kantorovich算子的导数和光滑模之间的关系及它们的线性组合的逼近等价定理.     

Jackson inequality for Banach spaces on the sphere

The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli of smoothness. The treatment applies to a wide class of Banach spaces of functions.      

On the complexity of the approximate table representation of discrete analogs of functions of finite smoothness in the metric ofL P

For discrete analogs of classes of functions of finite smoothness, we study the quantity log Approx characterizing the minimal necessary length of tables that allow us to reconstruct functions from these classes with error not exceeding 1 in the metric of the spaceL ^p . Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 643–647, November, 1998. The author wishes to express his gratitude to his scientific adviser O. B. Lupanov for setting the problem and attention to the author's work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01068.     

Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials

We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness wФ^r（f, t） where Ф is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.