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.
@article{bwmeta1.element.bwnjournal-article-apmv63z1p35bwm,
author = {A. Zeriahi},
title = {Approximation polynomiale et extension holomorphe avec croissance sur une vari\'et\'e alg\'ebrique},
journal = {Annales Polonici Mathematici},
volume = {63},
year = {1996},
pages = {35-50},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv63z1p35bwm}
}
A. Zeriahi. Approximation polynomiale et extension holomorphe avec croissance sur une variété algébrique. Annales Polonici Mathematici, Tome 63 (1996) pp. 35-50. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv63z1p35bwm/
[000] [B-T] E. Bedford and B. A. Taylor, Plurisubharmonic functions with logarithmic singularities, Ann. Inst. Fourier (Grenoble) 38 (4) (1988), 133-171. | Zbl 0626.32022
[001] [Be] S. Bernstein, Sur l'ordre de la meilleure approximation polynomiale des fonctions continues, Bruxelles, 1912.
[002] [Bj] J.-E. Björk, On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in , Ann. Inst. Fourier (Grenoble) 24 (4) (1974), 157-165. | Zbl 0288.32014
[003] [De] J.-P. Demailly, Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France 19 (1985). | Zbl 0579.32012
[004] [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.
[005] [H] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, 1973. | Zbl 0271.32001
[006] [L-G] P. Lelong and L. Gruman, Entire Functions of Several Variables, Springer, 1986. | Zbl 0583.32001
[007] [Ng] T. V. Nguyen, Croissance et meilleure approximation polynomiale des fonctions entières, Ann. Polon. Math. 24 (1972), 325-333. | Zbl 0241.30043
[008] [No] K. J. Nowak, The extension of holomorphic functions of polynomial growth on algebraic sets in , Osnabrücker Schriften zur Math., 1984.
[009] [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.
[010] [R,2] L. L. Ronkin, Entire functions, in: Encyclopedia of Math. Sci., Several Complex Variables III, Springer, 1986, 1-30.
[011] [R-W] K. Rusek and T. Winiarski, Criteria for regularity of holomorphic mappings, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 471-475. | Zbl 0507.14003
[012] [Sa] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1980), 493-502. | Zbl 0582.32023
[013] [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. | Zbl 0111.08102
[014] [Si,2] J. Siciak, Extremal plurisubharmonic functions in , Ann. Polon. Math. 39 (1981), 175-211. | Zbl 0477.32018
[015] [Si,3] J. Siciak, Approximation by transcendental polynomials, Ann. Polon. Math. 46 (1985), 299-309. | Zbl 0604.32013
[016] [Sk] H. Skoda, Morphismes surjectifs et fibrés linéaires semi-positifs, dans : Séminaire P. Lelong-H. Skoda (Analyse), Lecture Notes in Math. 694, Springer, 1978.
[017] [Wa] J. Walsh, Interpolation and Approximation by Rational Functions, Boston, 1960.
[018] [Wi,1] T. Winiarski, Approximation and interpolation of entire functions, Ann. Polon. Math. 23 (1970), 259-273. | Zbl 0205.37905
[019] [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. | Zbl 0257.32008
[020] [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).
[021] [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. | Zbl 0596.32025
[022] [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.
[023] [Ze,3] A. Zeriahi, Ensembles pluripolaires exceptionnels pour la croissance partielle des fonctions holomorphes, Ann. Polon. Math. 50 (1989), 81-91. | Zbl 0688.32004