关于Bernstein-Kantorovich算子的强逆不等式

Strong Converse Inequality for Bernstein-Kantorovich Operators

定义了一种新的K-泛函:K(f,t)∞n=infg∈C2[0,1]{‖f-g‖n∞+t‖δ2ng″‖∞n+t‖g′‖n∞},其中‖f‖n∞=supx∈[0,1]|δn-β(x)f(x)|,0≤β≤2,δ2n(x)=φ2(x)+1n,φ(x)=x(1-x).利用此K-泛函给出了Bernstein-Kantorovich算子点态逼近的强逆不等式,即若f∈C[0,1],β=α(1-λ),0<α≤2,0≤λ≤1,则x∈[0,1],及h∈(0,41),都存在正整数n及m满足|2hφλf(x)|≤Chαnα/2{‖Knf-f‖∞n+‖Kmnf-f‖∞n}.

A new kind of K-functional：K（f,t）∞^n=infg∈C2[0,1]{‖f-g‖∞^n＋t‖δ^2ng″‖∞^n＋t‖g′‖∞^n},is defined,where‖f‖∞^n=supx∈[0,1]｜δn^-β（x）f（x）｜,0≤β≤2,δn^2（x）=φ^2（x）＋1/n,φ（x）=x（1-x）.with the help of K（f,t）∞^n,the strong converse inequality on pointwise approximation by Bernstein-Kantorovich operators. Let f∈C[0,1],β=α（1-λ）,0〈α≤2,0≤λ≤1,then A↓x∈[0,1],and A↓h∈（0,1/4）,there exist two positive integers n and m satisfying ｜△↓^2hφ^λf（x）｜≤Ch^αn^α/2{‖Knf-f‖∞^n＋‖Kmnf-f‖∞^n}.