Interpolatory pointwise estimates for monotone polynomial approximation

TL;DR

This paper advances the understanding of monotone polynomial approximation by establishing interpolatory pointwise error estimates based on the second modulus of smoothness, resolving an open problem for functions with smoothness index  since 1985.

Contribution

It proves that for functions with certain smoothness, interpolatory estimates hold for large degrees, extending previous results and solving an open problem for  with  .

Findings

Interpolatory estimates valid for functions in Lip^* for sufficiently large degree n.

Established equivalence between nondecreasing Lip^* functions and existence of nondecreasing polynomial approximations with specific error bounds.

Resolved an open problem from 1985 regarding approximation error estimates for  functions.

Abstract

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 \[ |f(x)-P_n(x)| \leq C \left(\frac{\sqrt{1-x^2}}{n}\right)^\alpha, \quad x\in [-1,1]. \]