L_p空间中Post-Widder算子带权同时逼近的强逆不等式

Strong converse inequalities for weighted simultaneous approximation by Post-Widder operators in the space L_p

对于Post-Widder算子Pn(f,x),证明了当s∈N0=N∪{0},wf(s)∈Lp(0,∞)(1<p≤∞)时,存在某一正数m,使得ω2≤φf(s),1nw,pC(‖w(P(s)nf-f(s))‖p+‖w(P(s)mnf-f(s))‖p+1n‖wf(s)‖p),其中φ(x)=x;w(x)=xa(1+x)b;a,b∈R1;C>0;ω2φ(f,t)w,p是带权光滑模.

For Post-Widder operators Pn(f,x),s∈N0=N∪｛0｝ ,wf(s)∈Lp(0,∞)(1<p≤∞)and for some m,one can haveω2φf(s),1nw,p≤C(‖w(P(s)nf-f(s))‖p+‖w(P(s)mnf-f(s))‖p+1n‖wf(s)‖p),where φ(x)=x,w(x)=xa(1+x)b(a,b∈R1),C>0,ω2φ(f,t)w,pis the weighted modulus of smoothness.