THE NORM ESTIMATES FOR THE q-BERNSTEIN OPERATOR IN THE CASE q > 1

被引:8
作者
Wang, Heping [1 ]
Ostrovska, Sofiya [2 ]
机构
[1] Capital Normal Univ, Sch Math Sci, Beijing 100048, Peoples R China
[2] Atilim Univ, Dept Math, TR-06836 Ankara, Turkey
基金
北京市自然科学基金; 中国国家自然科学基金;
关键词
q-integers; q-binomial coefficients; q-Bernstein polynomials; q-Bernstein operator; operator norm; strong asymptotic order; POLYNOMIALS; CONVERGENCE; SATURATION; THEOREM;
D O I
10.1090/S0025-5718-09-02273-X
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The q-Bernstein basis with 0 < q < 1 emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on [0, 1]. In the case q > 1, the behavior of the q-Bernstein basic polynomials on [0, 1] combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of q-Bernstein polynomials in the case of q > 1. The aim of this paper is to present norm estimates in C[0, 1] for the q-Bernstein basic polynomials and the q-Bernstein operator B-n,B-q in the case q > 1. While for 0 < q <= 1, parallel to B-n,B-q parallel to = 1 for all n is an element of N, in the case q > 1, the norm parallel to B-n,B-q parallel to increases rather rapidly as n -> infinity. We prove here that parallel to B-n,B-q parallel to similar to C(q)q(n(n-1)/2)/n, n -> infinity with C-q = 2 (q(-2); q(-2))(infinity)/e. Such a fast growth of norms provides an explanation for the unpredictable behavior of q-Bernstein polynomials (q > 1) with respect to convergence.
引用
收藏
页码:353 / 363
页数:11
相关论文
共 22 条
[1]  
Andrews George E, 1999, Encyclopedia of Mathematics and its Applications, V71, DOI DOI 10.1017/CBO9781107325937
[2]  
[Anonymous], 2005, ADV STUD CONT MATH
[3]   THE QUANTUM GROUP SUQ(2) AND A Q-ANALOGUE OF THE BOSON OPERATORS [J].
BIEDENHARN, LC .
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1989, 22 (18) :L873-L878
[4]   On de Casteljau's algorithm [J].
Boehm, W ;
Müller, A .
COMPUTER AIDED GEOMETRIC DESIGN, 1999, 16 (07) :587-605
[5]  
Castellani L., 1996, Quantum Groups and Their Applications in Physics
[6]  
CHARALAMBIDES CA, J STAT PLAN IN PRESS
[7]  
GONSKA H, 1998, AUTOMAT COMPUT APPL, V7, P38
[8]   Convergence of generalized Bernstein polynomials [J].
Il'inskii, A ;
Ostrovska, S .
JOURNAL OF APPROXIMATION THEORY, 2002, 116 (01) :100-112
[9]  
ILINSKII A, 2004, MAT FIZ ANAL GEOM, V11, P434
[10]  
JING SC, 1994, J PHYS A-MATH GEN, V27, P493, DOI 10.1088/0305-4470/27/2/031