Preservation of Lipschitz constants by Bernstein type operators (Q812818)

Let \(\Delta _h^sf (x)\) be the forward difference of order \(s\) of a function \(f: A\subseteq R\rightarrow R \) and let \(\theta _\mu^s(t)= \sup _{\delta>0}\sup _{0<h\leq \delta;x,x+sh\in A}\{h^{-\mu}| \Delta _h^sf (x) | \}\) be higher order Lipschitz constant of \(f\). By \(\text{Lip}_\mu ^{(s)}A\) is denoted the space of continuous functions on \(A\), for which \(\theta _\mu ^s(t)<\infty \). The problem of the preservation of higher order Lipschitz constants in simultaneous approximation processes for certain classes of Bernstein type linear positive operators is studied. Namely, for such operators \(L_n\), it is proved that \(D^rf\in \text{Lip} _\mu ^{(s)}A\) implies that \(D^rL_n(f)\in \text{Lip} _\mu ^{(s)}A\), \( 0<\mu \leq 1 \) (\(D^r \) denotes the derivative of order \(r\)). Moreover, constants \(c>0\), sharp in certain cases, for preservation inequalities of the type \(\theta _\mu ^{(s)}(D^rL_nf)\leq c\theta _\mu ^{(s)}(D^rf)\) are found.