Jackson's inequality in the complex plane and the Lojasiewicz-Siciak inequality of Green's function

TL;DR

This paper generalizes Jackson's inequality for compact sets in the complex plane by incorporating Green's function bounds and establishes the Lojasiewicz-Siciak inequality as a necessary condition for this generalization.

Contribution

It extends Jackson's inequality to complex sets with Green's function bounds and links the Lojasiewicz-Siciak inequality as a necessary condition.

Findings

Jackson's inequality is generalized to complex plane sets with Green's function bounds.

The Lojasiewicz-Siciak inequality is shown to be necessary for the generalized Jackson inequality.

The paper highlights the importance of the Holder Continuity Property in approximation theory.

Abstract

We prove a generalization of Dunham Jackson's famous approximation inequality to the case of compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e. the well known Holder Continuity Property (HCP) and the less known but crucial Lojasiewicz-Siciak inequality (LS). Moreover, we show that (LS) is a necessary condition for our Jackson type inequality.