Durrmeyer-Bernstein算子的L_p饱和等价定理

L_p-saturation Theorem for Durrmeyer-Bernstein Operators

讨论了DurrmeyerBernstein算子Dn(f,x)在Lp空间的饱和问题.在处理线性算子逼近的饱和问题时,通常采用“抛物线技巧、Fourier技巧和积分方程技巧”,本文引入双线性泛函:An(f,φ)=(n+1)∫10(Dn(f,x)-f(x))φ(x)dx,利用积分方程技巧得出该算子在Lp[0,1]关于阶{1/n}和平凡类T={f｜f=const(a.e)}是Lp饱和的.

The saturation problem of the Durrmeyer-Bernstein operator D_n(f,x) in L_p space is discussed. To deal with the saturation problem of approximation of linear operators, parabola skill, Fourier skill and integral equation skill are usually used. In this paper, a twin-line-function is introduced as follows:A_n(f,φ)=(n+1)∫1_0(D_n(f,x)-f(x))φ(x)dx. By using the integral equation skill, the saturation order of the Durrmeyer-Bernstein operator in L_p[0,1] is obtained with respect to the order {1/n} and trivial species T={f|f=const(a.e.)}.