Sets of uniqueness for uniform limits of polynomials in several complex variables

[摘要] We investigate the sets of uniform limits A((B) over bar (n)), A((D) over bar (I)) of polynomials on the closed unit ball (B) over bar (n) of C-n and on the cartesian product (D) over bar (I) where I is an arbitrary set, maybe finite, infinite denumerable or non-denumerable and (D) over bar is the closed unit disc in C. The class A((D) over bar (I)) contains exactly all functions f : (D) over bar (I) -> C continuous with respect to the product topology on (D) over bar (I) and separately holomorphic. We consider sets of uniqueness for A((D) over bar (I)) (respectively for A((B) over bar (n))) to be compact subsets K of T-I (respectively of partial derivative(B) over bar (n)) where T = partial derivative D is the unit circle. If K has positive measure then K is a set of uniqueness. The converse does not hold. Finally, we do a similar study when the uniform convergence is not meant with respect to the usual Euclidean metric in C, but with respect to the chordal metric chi on C boolean OR {infinity}. (C) 2015 Elsevier Inc. All rights reserved.

[发布日期] 2015-12-15

[关键词] Polydisc algebra;Infinite dimensional holomorphy;Polynomial approximation;Chordal distance