On sign changes in weighted polynomial \(L^1\)-approximation (Q1857400)

In this paper the authors deal with the asymptotic distribution of sign changes of the error function \(f-B_{n,1}(f)\), \(n=0,1,2,\dots\), where \(f\in L^1_m[-1,1]\) and \(B_{n,1}(f)\) denotes the best \(L^1\)-approximation with respect to \(\mathcal P_n\), a class of polynomials of degree atmost \(n\). The following theorems have been given analogous to the theorems of \textit{A. Kroo} and \textit{J. J. Swetits} [Constructive Approximation 8, 87-103 (1992; Zbl 0763.41023)] for the general weights: Theorem 1. For the distribution of the sign changes, we have \(D[\nu_{n,w}-\mu]\leq C_4\sqrt{\varepsilon_n(w)}\) for all \(n=0,1,2,\dots\) where \(C_4<0\) does not depend on \(n\). Theorem 2. Let \(-1=t_1\dots <t_m=1\) be fixed points, \(a_1,\dots ,a_m\) fixed numbers, and \(w\) a weight function that satisfies \[ w(x)\geq C_5 \prod^m_{\ell=i} |x-t_i|^{a_i} \] with \(C_5>0\). Then \[ D[\nu_{n,w}-\mu]\leq C_6\frac{(\log n)^2}n \] for all \(n=1,2,\dots\).