CONTENTS 1. Introduction................................................................................................................ 3 2. The extremal function.............................................................................................. 8 3. Some lemmas on polynomials............................................................................. 12 4. Category theorems in topological groups........................................................... 16 5. Best approximation in Banach spaces................................................................ 17 6. Quasianalytic and quasientire functions in the sense of Bernstein............... 21 7. Uniqueness property of quasianalytic functions................................................ 26 8. Uniqueness theorems for quasientire functions............................................... 31 9. Differentiability properties of quasianalytic functions........................................ 34 10. Rational approximation to quasianalytic and quasientire functions............ 39 11. Some remarks on superposition of quasianalytic functions......................... 44 12. A piecing-together problem.................................................................................. 48 13. Quasianalytic and quasientire functions in open subsets of $R^n$........... 53 14. Quasianalytio functions in the sense of Denjoy-Carleman........................... 56 References.................................................................................................................... 64 List of symbols.............................................................................................................. 66
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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.
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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 of semialgebraic sets or other "small" subsets of ℝⁿ (ℂⁿ).
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We show that in the class of compact, piecewise $C^1$ curves K in $ℝ^n$, 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.
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We give an estimate of Siciak's extremal function for compact subsets of algebraic varieties in $ℂ^n$ (resp. $ℝ^n$). As an application we obtain Bernstein-Walsh and tangential Markov type inequalities for (the traces of) polynomials on algebraic sets.
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We show that in the class of compact sets K in $ℝ^n$ 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.