Interpolatory pointwise estimates for polynomial approximation (Q1584551)

Consider \(f\) to be an \(r\)-times continuously differentiable real-valued function on \([0,1]\), and let \(\omega_k\) denote the \(k\)th modulus of smoothness. An estimate of the form \[ |f(x)- p_n(x)|\leq c(r,k) \delta^r_n(x) \omega_k(f^{(k)}, \delta_n(x))\qquad (x\in [0,1]), \] where \(p_n\) is a polynomial of degree \(\leq n\), \(\delta_n(x)= \sqrt{x(1- x)}/n\), and where \(c(r, k)\) is a constant independent of \(f\) and \(n\), is called an interpolatory pointwise estimate. The authors give a fairly complete solution to the problem whether or not for a given pair \((k,r)\) such an estimate is valid. They emphasize on constrained, i.e. positive and monotone, approximation. However, also for the unconstrained case they strengthen some existing results.