On certain orthogonal polynomials, Nikolski- and Turán-type inequalities, and interpolatory properties of best approximants
Let \(B_{n,p}(f)\) be the best \(L_ p\)-approximant of \(f\in C[-1,1]\) by polynomials of degree at most \(n\) (\(1<p\leq\infty\)). The main result of the paper: Assume that for a given subsequence \((n_ j)\), \(n_ 1<n_ 2<\dots\), such that \(f=B_{n_ j,p}(f)\) is zero-free on some interval \((a,b)\subseteq[-1,1]\). Then \(\limsup_{j\to\infty} n_{j+1}/n_ j>1\). This result constitutes a positive answer to a conjecture of \textit{E. B. Saff} and \textit{B. Shekhtman} [Indag. Math., New Ser. 1, No. 4, 489-498 (1990; Zbl 0723.41022)] concerning the lacunarity of bad subsequences of best approximation polynomials. For this purpose the author obtains general oscillatory properties of orthogonal polynomials (of independent interest) and fine asymptotic estimates.
scientific article; zbMATH DE number 205434