Szasz-Durrmeyer算子逼近的强逆不等式

Strong-converse inequality of approximation for Szasz-Durrmeyer operators

引入新的K-泛函K(f,t)β研究Szasz-Durrmeyer算子逼近的强逆不等式,从而得到了算子逼近的特征刻画.1)设f∈CB[0,∞),则存在常数R>1,当l≥Rn时,有K(f,1/n)β≤Cln.(‖Mnf-f‖β+‖Mlf-f‖β);2)设0<h<1/16,0<α≤2,0≤λ≤1,则对每个x∈[0,∞),存在n=n(x,h)及正整数m,满足△2hφλf(x)≤Chαnα/2(‖Mnf-f‖β+‖Mmnf-f‖β).

This paper studies strong converse inequality of approximation for Szasz-Durrmeyer by introducing a new K-functional K（f, t）β. From these theorems, the characterization of approximation for these operators is derived： 1） For f∈CB [0, ∞）, there exists a constant R〉1 such that K（f,1/n）β≤C l/n·（‖Mn f-f‖β＋‖Ml-f‖β） for 1≥Rn,2）0〈h〈1/16,0〈α≤2,0≤λ≤1 ,there exist n=n（x,h） and a integer number m such that ｜△^2hψλf（x）｜≤Ch^αn^α/2（‖Mn f-f‖＋‖Mmn f-f‖β）.