INDICATORS OF GROWTH OF POLYNOMIALS OF BEST UNIFORM APPROXIMATION TO HOLOMORPHIC-FUNCTIONS ON COMPACTA IN C(N)

Let E be a compact and L-regular subset of C(N). Siciak has shown that a function f 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 su p(n --> infinity) (sup(E)\f - p(n)\1/n) less-than-or-equal-to 1/R (R > 1), where p(n) is a best approximating polynomial to f of degree not greater than n. The aim of this paper is to show that f has a holomorp hic extension to E(R) if for some sequence {p(n)} of the polynomials o f best approximation to f [GRAPHICS] and if f has such an extension, f or all {p(n)}, there holds [GRAPHICS] Here parallel p(n) parallel deno tes 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. (C) 1994 Academic Press, In c.