The equivalence of Shepard operators in \(L^p\) for \(p\geqq 1\) (Q2757213)

For \(f\in L^p_{[0,1]}\), \(1\leq p<\infty\), put NEWLINE\[NEWLINEL_{n,\lambda} (f,x)= (n+1)\sum^n_{k=0} r_k(x)\int^{(k+1)/(n+1)}_{k/(n+1)}f(u)du, \;\lambda> 1,NEWLINE\]NEWLINE where NEWLINE\[NEWLINEr_k(x)= {|x-k/n|^{-\lambda}\over\sum^n_{j=0}|x-j/n |^{-\lambda}}.NEWLINE\]NEWLINE The authors prove the theorem:NEWLINENEWLINENEWLINELet \(f(x)\in L^p_{[0,1]}\), \(p\geq 1\), \(0<\alpha<1\), and \(\lambda>3\). Then, the following conditions are equivalent: (A) \(\|L_{n,\lambda} (f)-f\|_{L^p} \sim n^{-\alpha}\); (B) \(\omega (f,\delta)_{L^p} \sim\delta^\alpha\). The proof (9 pages) uses ingenious calculations.