More quantitative results on walsh equiconvergence: I. Lagrange Case

Abstract

Letf∈A _ρ (ρ>1), whereA _ρ denotes the class of functions analytic in ¦z¦ <ρ but not in ¦z¦≤ρ. For any positive integerl, the quantity Δ _l,n−1(f; z) (see (2.3)) has been studied extensively. Recently, V. Totik has obtained some quantitative estimates for$\overline {\lim _{n \to \infty } } \max _{\left| z \right| = R} \left| {\Delta _{l,n - 1}^ - \left( {f;z} \right)} \right|^{1/n}$. Here we investigate the order of pointwise convergence (or divergence) of Δ _l,n−1(f; z), i.e., we study$B_1 \left( {f;z} \right) = \overline {\lim _{n \to \infty } } \left| {\Delta _{l,n - 1} \left( {f;z} \right)} \right|^{1/n}$. We also study some problems arising from the results of Totik.