Coconvex polynomial approximation
Let \(f\in\mathbb{C}[-1,1]\) change its convexity finitely many times, in the interval. The authors are interested in estimating the degree of approximation of \(f\) by polynomials, and by piecewise polynomials, which are coconvex with it, namely, polynomials and piecewise polynomials that change their convexity exactly at the points where \(f\) does. The authors obtain Jackson-type estimates and summarize the positive and negative results in a truth-table. They prove two theorems. Theorem 4.1. If \(f\in\Delta^2(Y_s)\cap\mathbb{C}^2\), then for each \(k\leq 3\), \[ E^{(2)}_n(f,Y_s)\leq\frac c{n^2}\omega^\varphi_k(f'',1/n) \leq\frac c{n^2}\omega_k(f'',1/n),\quad n\geq N , \] where \(c=c(s)\) and \(N=N(Y_s)\). Furthermore, if \(s=1\) and \(k\leq 2\), then \(N=k+1\). Theorem 4.2. Let \(r\geq 3\) and assume that \(f\in\Delta^2(Y_s)\cap\mathbb{C}^r\). Then \[ E^{(2)}_n(f,Y_s)\leq\frac c{n^r}\omega^\varphi_k(f^{(r)},1/n),\quad n\geq N , \tag{1} \] holds for each \(k\geq 1\), with constants \(c=c(k,r,s)\) and \(N=N(k,r,Y_s)\). Furthermore, if \(s=1\), then \((1)\) holds with \(N=k+r-1\).
scientific article; zbMATH DE number 1915646