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

Let E be a compact and L-regular subset of CN. Siciak has shown that a function ƒ on E has a holomoprhic extension to ER—the interior of the level curve of the Siciak extremal function—if and only if lim supn → ∞ (supE |ƒ − pn|1/n) ≤ 1/R (R > 1), where pn 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 ER if for some sequence {pn} of the polynomials of best approximation to ƒ [formula] and if ƒ has such an extension, for all {pn}, there holds [formula]. Here [formula] denotes a norm on the homogeneous terms of degree n in pn and cm(E), d(E) are some multidimensional counterparts of the logarithmic capacity and the Chebyshev constant, respectively.