ω,q-Bernstein多项式的Voronovskaya-型公式

Voronovskaya-Type Formulas forω,q-Bernstein Polynomials

讨论了ω,q-Bernstein多项式的Voronovskaya-型公式及其收敛的饱和性.给出了当0<q<1,0≤ω≤1,f∈C^1[0,1]时ω,q-Bernstein多项式的Voronovskaya-型公式.如果0<ω,q<1,f∈C^1[0,1],则ω,q-Bernstein多项式的收敛阶为o(q^n)当且仅当((f(1-q^(k-1)-f(1-q)~k))/((1-q^(k-1)-(1-q^k)))=f'(1-q^k),k=1,2,…还证明f如果f在[0,1]是凸的或者在(-ε,1+ε)(ε>0)解析,则ω,q-Bernstein多项式的收敛阶为o(q^n)当且仅当f是线性函数.

We discuss Voronovskaya-type formulas and saturation of convergence for w,q-Bernstein polynomials.We give explicit formulas of Voronovskaya-type for w,q-Bernstein polynomials for 0 g 1,0≤w≤1.If 0g 1,we show that the rate of convergence for w,q-Bernstein polynomials is o（q^n） if and only if（f（1-q^（k-1）-f（1-q^k））/（（1-q^（k-1）-（1-q^k））= f＇（1 - q^k）,k = 1,2,....We also prove that if f is convex on[0,1]or analytic on （-ε,1 ＋ε） for someε 0,then the rate of convergence for w,q-Bernstein polynomials is o（q^n） if and only if f is linear.