Szász-Durrmeyer算子的点态逆结果(英文)

The pointwise converse results for Szász-Durrmeyer operators

用一个单调函数ω(t)为中介 ,利用 Szász- Durrmeyer算子导数的性质以及该算子的可换性和光滑模ωφλ(f ,t)为特点 ,得到以下点态逼近逆定理 :对于 f∈ C[0 ,+∞ ) ,0≤λ≤ 1,φ(x) =x ,δn(x) =φ(x) +1/ n ,若|f (x) - Sn(f ,x) |≤ Mω(n- 1 /2δ1 -λn (x) ) ,其中ω(t)≥ 0 , ω(ut)≤ C(u2 +1)ω(t) ,则对任意 t>0 ,有ω2φλ(f ,t)≤ Ct2 ∑0 <n≤ t- 1(n +1)ω(n- 1 ) +Ct2‖ f‖ ,ω1 (f ,t)≤ Ct ∑0 <n≤ t- 1ω(n- 2 -λ2 ) +Ct‖ f‖ .此结果推广了有关ωφ(f ,t)和ω(f ,t)

With an increasing function ω(t) and the properties of modulu s of smoothness ω φ λ (f,t),the following pointwise conve rse results for Szász-Durrmeyer operators are given by using the communication and the properties of derivative about Szász-Durrmeyer. For f∈C[0,+∞),0≤λ≤1,φ(x)=[KF(]x,δ n(x)=φ(x)+1/n , if |f(x)-S n(f,x)|≤Mω(n -1/2 δ 1-λ n(x)), where ω(t)≥0, ω(ut)≤C(u 2+1)ω(t), for any u>0,t>0. Then for any t>0 ω 2 φ λ (f,t)≤Ct 2∑0<n≤t -1 (n+1)ω(n -1 )+Ct 2‖f‖, ω 1(f,t)≤Ct∑0<n≤t -1 ω(n -2-λ2 )+Ct‖f‖, where ω 1(f,t)= sup 0<h≤t‖△ hf‖ C[0,∞) ,ω 2 φ λ (f,t)= sup [DD(X]|h|≤t‖△ 2 hφ λ f‖ C . It includes the converse results of ω φ(f,t) and ω(f,t).