Interpolatory pointwise estimates for monotone polynomial approximation

Given a nondecreasing function $f$ on $[-1,1]$, we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at $\pm 1$. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at $\pm 1$). We call such estimates "interpolatory estimates". In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness $\omega_2(f,\cdot)$ of $f$ evaluated at $\sqrt{1-x^2}/n$ and were valid for all $n\ge1$. The current paper is devoted to proving that if $f\in C^r[-1,1]$, $r\ge1$, then the interpolatory estimates are valid for the second modulus of smoothness of $f^{(r)}$, however, only for $n\ge N$ with $N= N(f,r)$, since it is known that such estimates are in general invalid with $N$ independent of $f$. Given a number $\alpha>0$, we write $\alpha=r+\beta$ where $r$ is a nonnegative integer and $0<\beta\le1$, and denote by $Lip^*\alpha$ the class of all functions $f$ on $[-1,1]$ such that $\omega_2(f^{(r)}, t) = O(t^\beta)$. Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for $\alpha\geq 2$ since 1985: If $\alpha>0$, then a function $f$ is nondecreasing and in $Lip^*\alpha$, if and only if, there exists a constant $C$ such that, for all sufficiently large $n$, there are nondecreasing polynomials $P_n$, of degree $n$, such that $$|f(x)-P_n(x)| \leq C \left(\frac{\sqrt{1-x^2}}{n}\right)^\alpha, \quad x\in [-1,1].$$