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Quasianalytic functions in the sense of Bernstein

100%
Pleśniak W.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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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Siciak's extremal function in complex and real analysis

95%
Pleśniak W.
Annales Polonici Mathematici
|
2003
|
tom 80
|
nr 1
37-46
EN
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.
3
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Multivariate polynomial inequalities viapluripotential theory and subanalytic geometry methods

95%
Pleśniak W.
Banach Center Publications
|
2006
|
tom 72
|
nr 1
251-261
EN
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).
4
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Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves

61%
Baran M., Pleśniak W.
Studia Mathematica
|
1997
|
tom 125
|
nr 1
83-96
EN
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.
5
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Polynomial inequalities on algebraic sets

61%
Baran M., Pleśniak W.
Studia Mathematica
|
2000
|
tom 141
|
nr 3
209-219
EN
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.
6
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Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities

61%
Baran M., Pleśniak W.
Studia Mathematica
|
2000
|
tom 141
|
nr 3
221-234
EN
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.
7
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Characterization of quasi-analytic functions of several variables by means of rational approximation

60%
Pleśniak W.
Annales Polonici Mathematici
|
1972-1973
|
tom 27
|
nr 2
149-157
8
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Bibliography of Zdzisław Opial compiled by Jan Malczak and Andrzej Pelczar

48%
Pleśniak W.
Annales Polonici Mathematici
|
1990
|
tom 51
|
nr 1
13-20
9
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A Cantor regular set which does not have Markov's property

42%
Pleśniak W.
Annales Polonici Mathematici
|
1990
|
tom 51
|
nr 1
269-274
10
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On superposition of quasianalytic functions

36%
Pleśniak W.
Annales Polonici Mathematici
|
1972
|
tom 26
|
nr 1
73-84
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