Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves

Baran, M. ; Pleśniak, W.

Studia Mathematica, Tome 122 (1997), p. 83-96 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1997-01-01

Zbl 0895.41011

EUDML-ID : urn:eudml:doc:216424

Bibliographie

[00000] [Ba1] M. Baran, Plurisubharmonic extremal function and complex foliations for the complement of convex sets in $ℝ^n$, Michigan Math. J. 39 (1992) 395-404.

[00001] [Ba2] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in $ℝ^n$, Proc. Amer. Math. Soc. 123 (1995) 485-494. | Zbl 0813.32011

[00002] [Ba3] M. Baran, Bernstein type theorems for compact sets in $ℝ^n$ revisited, J. Approx. Theory 79 (1994) 190-198. | Zbl 0819.41013

[00003] [Ba4] M. Baran, Markov inequality on sets with polynomial parametrization, Ann. Polon. Math. 60 (1994) 69-79. | Zbl 0824.41014

[00004] [BaPl] M. Baran and W. Pleśniak, Markov’s exponent of compact sets in $ℂ^n$, Proc. Amer. Math. Soc. 123 (9) (1995) 2785-2791. | Zbl 0841.41011

[00005] [BeRi] R. Benedetti and J.-J. Risler, Real Algebraic and Semi-Algebraic Sets, Hermann, Paris, 1990.

[00006] [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.

[00007] [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. | Zbl 0824.41015

[00008] [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. | Zbl 0834.41012

[00009] [BM1] 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, T. Rassias et al. (eds.), World Scientific, Singapore, 1992, 81-100.

[00010] [BM2] L. Bos and P. Milman, Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains, Geom. Funct. Anal. 5 (1995) 853-923. | Zbl 0848.46022

[00011] [Chir] E. M. Chirka, Complex Analytic Sets, Kluwer Acad. Publ., Dordrecht, 1989.

[00012] [CS1] J. G. van der Corput und G. Schaake, Ungleichungen für Polynome und trigonometrische Polynome, Compositio Math. 2 (1935) 321-361.

[00013] [CS2] J. G. van der Corput und G. Schaake, Berichtigung zu: Ungleichungen für Polynome und trigonometrische Polynome, ibid. 3 (1936) 128. | Zbl 0013.10803

[00014] [FeNa] C. Fefferman and R. Narasimhan, Bernstein's inequality on algebraic curves, Ann. Inst. Fourier (Grenoble) 43 (1993) 1319-1348. | Zbl 0842.26013

[00015] [Goe1] P. Goetgheluck, Inégalité de Markov dans les ensembles effilés, J. Approx. Theory 30 (1980) 149-154. | Zbl 0457.41015

[00016] [Goe2] P. Goetgheluck, Polynomial inequalities on general subsets of $ℝ^N$, Colloq. Math. 57 (1989) 127-136. | Zbl 0698.26004

[00017] [Gon] A. Goncharov, A compact set without Markov’s property but with an extension operator for $C^∞$-functions, Studia Math. 119 (1996) 27-35. | Zbl 0857.46013

[00018] [Har] J. Harris, Algebraic Geometry, Springer, New York, 1992.

[00019] [Hart] R. Hartshorne, Algebraic Geometry, Springer, New York, 1987.

[00020] [Jon1] A. Jonsson, Markov's inequality on compact sets, in: Orthogonal Polynomials and Their Applications, C. Brezinski, L. Gori and A. Ronveaux (eds.), J. G. Bultzer AG, 1991, 309-313.

[00021] [Jon2] A. Jonsson, Markov's inequality and zeros of orthogonal polynomials on fractal sets, J. Approx. Theory 78 (1994) 87-97. | Zbl 0918.41014

[00022] [Jos] B. Josefson, On the equivalence between locally and globally polar sets for plurisubharmonic functions in $ℂ^n$, Ark. Mat. 16 (1978) 109-115. | Zbl 0383.31003

[00023] [K] M. Klimek, Pluripotential Theory, Oxford Univ. Press, London, 1991.

[00024] [L] G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966. | Zbl 0153.38901

[00025] [Ł] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel, 1991. | Zbl 0747.32001

[00026] [MiRa] G. V. Milanović and T. M. Rassias, On Markov-Duffin-Schaeffer inequalities, J. Nat. Geometry 5 (1994) 29-41. | Zbl 0788.26011

[00027] [PP1] W. Pawłucki and W. Pleśniak, Markov’s inequality and $C^∞$ functions on sets with polynomial cusps, Math. Ann. 275 (1986) 467-480. | Zbl 0579.32020

[00028] [PP2] W. Pawłucki and W. Pleśniak, Extension of $C^∞$ functions from sets with polynomial cusps, Studia Math. 88 (1988) 279-287. | Zbl 0778.26010

[00029] [Pl1] W. Pleśniak, On superposition of quasianalytic functions, Ann. Polon. Math. 26 (1972) 73-84. | Zbl 0207.08101

[00030] [Pl2] W. Pleśniak, Remarques sur une généralisation de l'inégalité de S. Bernstein, C. R. Acad. Sci. Paris Sér. A 284 (1977) 1211-1213. | Zbl 0357.32010

[00031] [Pl3] W. Pleśniak, Markov’s inequality and the existence of an extension operator for $C^∞$ functions, J. Approx. Theory 61 (1990) 106-117.

[00032] [Sa] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1983), 493-502. | Zbl 0582.32023

[00033] [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. | Zbl 0111.08102

[00034] [Si2] J. Siciak, Extremal plurisubharmonic functions in $ℂ^n$, Ann. Polon. Math. 39 (1981) 175-211. | Zbl 0477.32018

[00035] [Si3] J. Siciak, Rapid polynomial approximation on compact sets in $ℂ^n$, Univ. Iagell. Acta Math. 30 (1993) 145-154. | Zbl 0838.32007

[00036] [Sin] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer, Berlin, 1970.

[00037] [Z] A. Zeriahi, Inégalités de Markov et développement en série de polynômes orthogonaux des fonctions $C^∞$ et $A^∞$, in: Proc. Special Year of Complex Analysis of the Mittag-Leffler Institute 1987-88, J. F. Fornaess (ed.), Princeton Univ. Press, Princeton, N.J., 1993, 693-701.