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2000 | 141 | 3 | 221-234
Tytuł artykułu

Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
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.
Czasopismo
Rocznik
Tom
141
Numer
3
Strony
221-234
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-11-23
poprawiono
2000-04-03
Twórcy
autor
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
autor
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Bibliografia
  • [Ba1] M. Baran, Siciak's extremal function of convex sets in $ℂ^n$, Ann. Polon. Math. 48 (1988), 275-280.
  • [Ba2] M. Baran, Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in $ℝ^n$, Michigan Math. J. 39 (1992), 395-404.
  • [Ba3] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in $ℝ^n$, Proc. Amer. Math. Soc. 123 (1995), 485-494.
  • [Ba4] M. Baran, Bernstein type theorems for compact sets in $ℝ^n$ revisited, J. Approx. Theory 79 (1994), 190-198.
  • [Ba5] M. Baran, Markov inequality on sets with polynomial parametrization, Ann. Polon. Math. 60 (1994), 69-79.
  • [BaPl1] M. Baran and W. Pleśniak, Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves, Studia Math. 125 (1997), 83-96.
  • [BaPl2] M. Baran and W. Pleśniak, Polynomial inequalities on algebraic sets, this issue, 209-219.
  • [BeT] E. Bedford and B. A. Taylor, The complex equilibrium measure of a symmetric convex set in $ℝ^n$, Trans. Amer. Math. Soc. 294 (1986), 705-717.
  • [BeRi] R. Benedetti and J.-J. Risler, Real Algebraic and Semi-Algebraic Sets, Hermann, Paris, 1990.
  • [Bern] S. N. Bernstein, Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné, Mém. Acad. Roy. Belg. 4 (2) (1912), 1-103.
  • [BLMT] L. Bos, N. Levenberg, P. Milman and B. A. Taylor, Tangential Markov inequalities characterize algebraic submanifolds of $ℝ^N$, Indiana Univ. Math. J. 44 (1995), 115-138.
  • [BLT] L. Bos, N. Levenberg and B. A. Taylor, Characterization of smooth, compact algebraic curves in $ℝ^2$, in: Topics in Complex Analysis, P. Jakóbczak and W. Pleśniak (eds.), Banach Center Publ. 31, Inst. Math. Polish Acad. Sci., Warszawa, 1995, 125-134.
  • [BoMi1] L. Bos and P. Milman, On Markov and Sobolev type inequalities on compact subsets in $ℝ^n$, in: Topics in Polynomials in One and Several Variables and Their Applications, Th. Rassias et al. (eds.), World Scientific, Singapore, 1992, 81-100.
  • [BoMi2] L. Bos and P. Milman, Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains, Geom. Funct. Anal. 5 (1995), 853-923.
  • [Bru] A. Brudnyi, A Bernstein-type inequality for algebraic functions, Indiana Univ. Math. J. 46 (1997), 93-116.
  • [CS1] J. G. van der Corput und G. Schaake, Ungleichungen für Polynome und trigonometrische Polynome, Compositio Math. 2 (1935), 321-361.
  • [CS2] J. G. van der Corput und G. Schaake, Berichtigung zu: Ungleichungen für Polynome und trigonometrische Polynome, ibid. 3 (1936), 128.
  • [DŁS] Z. Denkowska, S. Łojasiewicz et S. Stasica, Certaines propriétés élémentaires des ensembles sous-analytiques, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 529-536.
  • [FeNa1] C. Fefferman and R. Narasimhan, Bernstein's inequality on algebraic curves, Ann. Inst. Fourier (Grenoble) 43 (1993), 1319-1348.
  • [FeNa2] C. Fefferman and R. Narasimhan, On the polynomial-like behavior of certain algebraic functions, ibid. 44 (1994), 1091-1179.
  • [FeNa3] C. Fefferman and R. Narasimhan, A local Bernstein inequality on real algebraic varieties, Math. Z. 223 (1996), 673-692.
  • [Hir] H. Hironaka, Introduction to Real-Analytic Sets and Real-Analytic Maps, Istituto Matematico 'L. Tonelli', Pisa, 1973.
  • [Jos] B. Josefson, On the equivalence between locally and globally polar sets for plurisubharmonic functions in $ℂ^n$, Ark. Mat. 16 (1978), 109-115.
  • [K] M. Klimek, Pluripotential Theory, Oxford Univ. Press, London, 1991.
  • [Lu] M. Lundin, The extremal plurisuhbarmonic function for the complement of convex subsets of $ℝ^N$, Michigan Math. J. 32 (1985), 196-201.
  • [PaPl1] W. Pawłucki and W. Pleśniak, Markov's inequality and $C^∞$ functions on sets with polynomial cusps, Math. Ann. 275 (1986), 467-480.
  • [PaPl2] W. Pawłucki and W. Pleśniak, Extension of $C^∞$ functions from sets with polynomial cusps, Studia Math. 88 (1988), 279-287.
  • [Pl1] W. Pleśniak, Markov's inequality and the existence of an extension operator for $C^∞$ functions, J. Approx. Theory 61 (1990), 106-117.
  • [Pl2] W. Pleśniak, Recent progress in multivariate Markov inequality, in: Approximation Theory (In Memory of A. K. Varma), N. K. Govil et al. (eds.), Pure Appl. Math. 212, Marcel Dekker, New York, 1998, 449-464.
  • [RoYo] N. Roytwarf and Y. Yomdin, Bernstein classes, Ann. Inst. Fourier (Grenoble) 47 (1997), 825-858.
  • [Sa] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1983), 493-502.
  • [Si1] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357.
  • [Si2] J. Siciak, Extremal plurisubharmonic functions in $ℂ^n$, Ann. Polon. Math. 39 (1981), 175-211.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv141i3p221bwm
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