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1996 | 63 | 1 | 35-50
Tytuł artykułu

Approximation polynomiale et extension holomorphe avec croissance sur une variété algébrique

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FR
Abstrakty
EN
We first give a general growth version of the theorem of Bernstein-Walsh-Siciak concerning the rate of convergence of the best polynomial approximation of holomorphic functions on a polynomially convex compact subset of an affine algebraic manifold. This can be considered as a quantitative version of the well known approximation theorem of Oka-Weil. Then we give two applications of this theorem. The first one is a generalization to several variables of Winiarski's theorem relating the growth of an entire function to the rate of convergence of its best polynomial approximation; the second application concerns the extension with growth of an entire function from an algebraic submanifold to the whole space.
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autor
  • Laboratoire d'Analyse, Université Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse Cedex, France
Bibliografia
  • [B-T] E. Bedford and B. A. Taylor, Plurisubharmonic functions with logarithmic singularities, Ann. Inst. Fourier (Grenoble) 38 (4) (1988), 133-171.
  • [Be] S. Bernstein, Sur l'ordre de la meilleure approximation polynomiale des fonctions continues, Bruxelles, 1912.
  • [Bj] J.-E. Björk, On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in $ℂ^n$, Ann. Inst. Fourier (Grenoble) 24 (4) (1974), 157-165.
  • [De] J.-P. Demailly, Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France 19 (1985).
  • [D-M] P. B. Djakov and B. S. Mityagin, The structure of polynomial ideals in the algebra of entire functions, preprint of the Institute of Mathematics, Polish Academy of Sciences, no. 123, 1977.
  • [H] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, 1973.
  • [L-G] P. Lelong and L. Gruman, Entire Functions of Several Variables, Springer, 1986.
  • [Ng] T. V. Nguyen, Croissance et meilleure approximation polynomiale des fonctions entières, Ann. Polon. Math. 24 (1972), 325-333.
  • [No] K. J. Nowak, The extension of holomorphic functions of polynomial growth on algebraic sets in $ℂ^n$, Osnabrücker Schriften zur Math., 1984.
  • [R,1] L. L. Ronkin, Continuation with estimates of holomorphic functions on sets of zeros of polynomials, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 128, 36 (1981), 89-103.
  • [R,2] L. L. Ronkin, Entire functions, in: Encyclopedia of Math. Sci., Several Complex Variables III, Springer, 1986, 1-30.
  • [R-W] K. Rusek and T. Winiarski, Criteria for regularity of holomorphic mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 471-475.
  • [Sa] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1980), 493-502.
  • [Si,1] J. Siciak, On some extermal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357.
  • [Si,2] J. Siciak, Extremal plurisubharmonic functions in $ℂ^n$, Ann. Polon. Math. 39 (1981), 175-211.
  • [Si,3] J. Siciak, Approximation by transcendental polynomials, Ann. Polon. Math. 46 (1985), 299-309.
  • [Sk] H. Skoda, Morphismes surjectifs et fibrés linéaires semi-positifs, dans : Séminaire P. Lelong-H. Skoda (Analyse), Lecture Notes in Math. 694, Springer, 1978.
  • [Wa] J. Walsh, Interpolation and Approximation by Rational Functions, Boston, 1960.
  • [Wi,1] T. Winiarski, Approximation and interpolation of entire functions, Ann. Polon. Math. 23 (1970), 259-273.
  • [Wi,2] T. Winiarski, Application of approximation and interpolation methods to the examination of entire functions of n variables, Ann. Polon. Math. 28 (1973), 98-121.
  • [Za] V. P. Zakharyuta, Extremal plurisubharmonic functions, orthogonal polynomials and the Bernstein-Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math. 33 (1976), 137-148 (in Russian).
  • [Ze,1] A. Zeriahi, Meilleure approximation polynomiale et croissance des fonctions holomorphes sur une variété algébrique affine, Ann. Inst. Fourier (Grenoble) 37 (2) (1987), 79-104.
  • [Ze,2] A. Zeriahi, Fonction de Green pluricomplexe à pôle à l'infini sur un espace de Stein parabolique et applications, Math. Scand. 69 (1991), 89-126.
  • [Ze,3] A. Zeriahi, Ensembles pluripolaires exceptionnels pour la croissance partielle des fonctions holomorphes, Ann. Polon. Math. 50 (1989), 81-91.
Typ dokumentu
Bibliografia
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