Indicators of Growth of Polynomials of Best Uniform Approximation to Holomorphic Functions on Compacta in CN

Abstract

Let E be a compact and L-regular subset of C N. Siciak has shown that a function ƒ on E has a holomoprhic extension to E R—the interior of the level curve of the Siciak extremal function—if and only if lim sup n → ∞ (sup E |ƒ − p n | 1/ n ) ≤ 1/ R ( R > 1), where p n is a best approximating polynomial to ƒ of degree not greater than n. The aim of this paper is to show that ƒ has a holomorphic extension to E R if for some sequence { p n } of the polynomials of best approximation to ƒ [formula] and if ƒ has such an extension, for all { p n }, there holds [formula]. Here [formula] denotes a norm on the homogeneous terms of degree n in p n and c m ( E), d( E) are some multidimensional counterparts of the logarithmic capacity and the Chebyshev constant, respectively.