CONTENTS1. Introduction................................................................................................................ 32. The extremal function.............................................................................................. 83. Some lemmas on polynomials............................................................................. 124. Category theorems in topological groups........................................................... 165. Best approximation in Banach...
We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of ℝⁿ or ℂⁿ but also their versions for pieces...
The Siciak extremal function establishes an important link between polynomial approximation in several variables and pluripotential theory. This yields its numerous applications in complex and real analysis. Some of them can be found on a rich list drawn up by Klimek in his well-known monograph "Pluripotential Theory". The purpose of this paper is to supplement it by applications in constructive function theory.
We give an estimate of Siciak’s extremal function for compact subsets of algebraic varieties in (resp. ). As an application we obtain Bernstein-Walsh and tangential Markov type inequalities for (the traces of) polynomials on algebraic sets.
We show that in the class of compact, piecewise curves K in , the semialgebraic curves are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for the derivatives of (the traces of) polynomials on K.
We show that in the class of compact sets K in with an analytic parametrization of order m, the sets with Zariski dimension m are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on K.
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