Whitney theorem of interpolatory type for \(k\)-monotone functions (Q5937043)

The author proves the Whitney-type theorem for the class \(\Delta^k[a,b]\) of functions having nonnegative \(k\)-th divided differences for every choice of \(k+1\) distinct points: Let \(p_{m-1}(f)\) be the Lagrange polynomial of degree \(m-1\), \(m\leq k\), interpolating \(f\in\Delta^k[a,b]\cap L_p[a,b]\), \(0<p\leqslant\infty\), at \(m\) arbitrary points located in \(J_A=[a+A(b-a),b-A(b-a)]\), where \(0<A<1/2\). Then, \(\|f-p_{m-1}(f)\|_{L_p[a,b]}\leq C(A,k,p) \omega_m(f, b-a,[a,b])_p.\) Except for the case \(m=1\) and \(p=\infty\), this eatimate is no longer true if \(A=0\). The condition that \(J_A\) is ``in center'' of \([a,b]\) is essential, and cannot be removed. The result on simultaneous approximation of \(f\) and its derivatives \(f^{(i)}\) by \(p_{m-1}\) and \(p_{m-1}^{(i)}\) is also obtained.