tailieunhanh - Báo cáo " POLYNOMIAL APPROXIMATION ON POLYDISKS "

In this paper we give results about polynomial approximation on the closed polydisk in Cn . 1. Introduction Let X be a compact subset of Cn . By C(X) we denote the space of all continuous complex-valued functions on X, with norm f X = max{|f (z)| : z ∈ X}, and let P (X) denote the closure of set of polynomials in C(X). The polynomially convex hull of X will ˆ be denoted by X and difined by ˆ X = {z ∈ Cn : |p(z)| p X for every polynomial p}. | VNU. JOURNAL OF SCIENCE Mathematics - Physics. N03 - 2005 POLYNOMIAL APPROXIMATION ON POLYDISKS Kieu Phuong Chi Department of Mathematics Vinh University Abstract. In this paper we give results about polynomial approximation on the closed polydisk in Cn. 1. Introduction Let X be a compact subset of Cn. By C X we denote the space of all continuous complex-valued functions on X with norm Ilf x max f z z E X and let P X denote the closure of set of polynomials in C X . The polynomially convex hull of X will be denoted by X and difined by X z E Cn p z p X for every polynomial p . X is called polynomially convex if X X. One necessary condition for P X C X is X is polynomially convex. Let M be real manifold in Cn. We say that M has totally real if M has no complex tangent vectors . TaM n i TaM 0 . It is well-known that every continuous function on compact subset X of totally real manifold M is the uniform limit of sequence of polynomials . P X C X . Let D be small closed polydisk in Cn centered at the origin and f1 f 2 . fm E C D By z1 Z2 . zn f1 f2 . fm D we denote the function algebra consisting of uniform limits on D of all polynomials in zi z2 . zn and fl f2 . fm. The problem is that to find conditions of f1 . fm such that z1 z2 . zn fl f2 . fm D C D . Nguyen Quang Dieu . de Paepe . have many results if D is disk. Nguyen Quang Dieu and . de Paepe have shown that z f z D C D for some choices of f while for other choices of f to have z f z D C D see 4 5 6 7 . In the general H. Alexander and 1 Theorem used the result about approximation of totally real manifold to proved the following results. Theorem . Suppose R1 z . Rn z are complex-value continuoustly differentiable function on D satisfy conditions R z R z k z z Vz z E D where 0 k 1 R z R1 z . Rn z and w y w112 . wn 2 with w w1 . wn E Cn. Then zi . zn z1 R1 z . zn Rn z D C D . Typeset by AmS-TeX 10 Polynomial approximation on polydisks 11 The line of proof of Wermer is to .