Polynomial approximation of piecewise analytic functions
Let the function \(f\) be continuous on \([-1,1]\) and ``piecewise analytic'' there in some sense. Then \textit{K. G. Ivanov}, \textit{E. B. Saff} and \textit{V. Totik} [Proc. Am. Math. Soc. 106, No. 1, 153-161 (1989; Zbl 0668.41006)] and \textit{E. B. Saff} and \textit{V. Totik} [J. Lond. Math. Soc., II. Ser. 39, 487-498 (1989; Zbl 0683.41011)] proved the existence of polynomials \(p_n\) of degree \(n\) which are ``almost best approximating'' \(f\) uniformly on \([-1,1]\) and in addition admit a geometric convergence rate in points, where \(f\) itself is analytic. Here, the author gives another more constructive approach to results of this kind by projecting the problems onto the unit circle and by use of summability methods, and he improves some of the above results.
scientific article; zbMATH DE number 1002154