An extension of a theorem of Walsh
Let \(f(z)\in A_{\rho}\) \((\rho >1)\), the class of function analytic in \(| z| <\rho\) but not in \(| z| \leq \rho\). For a given positive integer r, we study the polynomial \(P_{n,r}(z)\) which minimizes \(\sum^{r-1}_{\nu =0}\sum^{m-1}_{k=0}| Q_ n^{(\nu)}(\omega^ k)-f^{(\nu)}(\omega^ k)|^ 2,\) over all polynomials \(Q_ n\in \pi_ n\), where \(\omega^ m=1\), \(m=nq+c\). This leads to an extension of a recent result of \textit{T. J. Rivlin} [J. Approximation Theory 36, 334-345 (1982; Zbl 0504.42016)] on Walsh equiconvergence. Some other extensions and results related to those of Saff and Varga are given.
scientific article; zbMATH DE number 3882952