摘要
本文主要给出Ｓｚａｓｚ－Ｍｉｒａｋｊａｎ算子逼近的一个下方估计，若ｗ（ｔ）∈Ｎβ（０＜β＜１），ｆ∈Ｃ１［０，∞）且ｗ（ｆ，δ）～ｗ（δ），则存在常数ｋ＞０，使‖Ｓｎ（ｆ）－ｆ‖Ｃ１≥ｋｗ（１ｎ）．
Suppose that C=C[0,∞), and that C 1[0,∞) be a set of Continuous and bounded functions on [0,∞), The norm of the function f(x) is defined by ‖f‖ C 1 = sup x∈[0,∞)|f(x)|, f∈C 1[0,∞) Let S n be Szasz-Mirakjan operator, i.e. S n(f,x)=e -nx ∑∞k=0f(kn)(nx) kk!. We have proved the following theorems. Theorem 1 Let w(t)∈N β(0<β<1), f∈C 1[0,∞) such that w(f,δ)～w(δ), then, there is a constant k>0 such that ‖S n(f)-f‖ C 1 ≥kw(1n). Theorem 2 Let w(t)∈N β(0<β<1) and ‖S n(f)-f‖ C 1 =0(w(1n)), then w(f,t)=0(w(t)). Theorm 3 Let 0≤α<β<1 and 0<k 5t β≤w(f,t)≤k 6t α them, ‖S n(f)-f‖ C 1 ≥kn -β(1-α)1-β >0. For general positive and linear operator L n(f,x), if it's satisfied |L n′(f,x)|≤Mnw(f,1n), then Theorem 1 hold true for the operator L n(f,x).
