An asymptotic result for the degree of approximation by monotone polynomials (Q1263020)

Let \(P_ n\) be the class of all nondecreasing polynomials of degree at most n defined on a real interval \(I=[a,b]\), \(\ell =b-a\), and \(E_ n(f)\) be the degree of approximation of f by polynomials in \(P_ n\). Analogous to the modulus of continuity \(\omega(f,.)\), define two nonnegative functions by \(\mu(\delta)=\sup \{f(y)-f(x):\) \(x,y\in I\), \(0\leq y-x\leq \delta \}\) and \({\bar \mu}(\delta)=\sup \{f(x)-f(y):\) \(x,y\in I\), \(0\leq y-x\leq \delta\}\), \(\delta\in [0,1]\), respectively. Let C denote the space of all real continuous functions and K the convex cone of real nondecreasing functions on I. The author studies the lower and upper bounds on the degree of approximation and an asymptotic result by establishing the following two theorems: Theorem 1. Let \(f\in C-K\), then for every positive integer m and for all \(n\geq m\), \[ 0\leq E_ n(f)--{\bar \mu}(f,1)\leq \pi \{\| f\| +- {\bar \mu}(f,1)\}\rho^{m+1}(n+1) \binom{n+1}{m+1}^{-1}, \] \(\rho =2l/\lambda \geq 2\), where \(\lambda= \sup \{\delta \in [0,1]:\) \(\omega(f,\delta)= {\bar \mu}(f,1)\}>0\). Theorem 2. Let \(f\in C-K\), then \[ 0\leq E_ n(f)- {\bar\mu}(f,1)\leq c_ 2n^{3/2}(\frac{\rho}{\rho+1})^ n, \] \(n\to \infty\) and \(c_ 2\) depends on f.