Distribution of extreme points of the error curve of best approximation by incomplete polynomials (Q1340504)

The paper is concerned with the distribution in \((-1, 1)\) of the extreme points of the error curve of the best uniform approximation of a function \(f\in C([-1,1 ])\), such that \(f(-1) =0\), by \(q\) in the set \(\Pi_{n,m}= \{(x+ 1)^n P(x)\): \(P\in \Pi_m\}\) of incomplete polynomials of type \((n,m)\). \(T_{n,m}\) denotes the constrained Chebyshev polynomial of \(\Pi_{n,m}\), that is if \(q_0\in \Pi_{n, m-1}\) and \[ |(x+ 1)^{n+m}- q_0 |= \inf\{|(x+ 1)^{n+m} -q|:\;q\in \Pi_{n, m}\}= E \] then \(T_{n,m} (x)= ((x+ 1)^{n+ m}- q_0)/E\). The extreme points of \(T_{n,m}\) are the \(m+1\) points \(-1\leq \xi_0^{(n, m)}< \cdots< \xi_m^{(n, m)}\leq 1\) such that \(T_{n,m} (\xi_j^{(n, m)} )=(- 1)^{m-n}\) for \(j= 0,\dots, m\). The first two theorems are concerned with the distribution in \([-1,1 ]\) of the extreme points and zeros of \(T_{n, m}\). The main result of the paper states that if \(n\neq 0\), \(f\in C([-1, 1])\), \(f(-1) =0\) and \(f\not\in \cup\{ \Pi_{n,m}\): \(m=1, 2, \dots\}\), if \(P_m\) is the best approximation to \(f\) from \(\Pi_{n,m}\) and \(-1\leq x_0< x_1< \cdots< x_{m+1}\leq 1\) are alternation points: \(P_m (x_j)- f(x_j)= \sigma (-1)^j |P_m- f|\) for \(j=0, \dots, m+1\), where \(\sigma= \pm 1\), and \(\Delta_m= \max_{0\leq k\leq m+1}|\cos^{-1} (x_k)- \cos^{-1} (\xi_k^{n, m+1}) |\), then \(\varliminf_{m\to \infty} {{\Delta_m m^{2/3}} \over {(\log m)^{2/3}}} \leq 3\). A Corollary to the theorem is an improvement of a result of \textit{M. I. Kadec} [Usp. Mat. Nauk 15, No. 1(91), 199-202 (1960; Zbl 0136.364)].