Polynomial approximation on compact sets of zero logarithmic capacity and analyticity of functions (Q1382394)

Let \(E_n(f,K)=\inf\{\sup_{z\in K}|f(z)-P_n(z)|:\deg P_n\leq n\}\) be the best polynomial approximation to a function \(f\) on a regular compact \(K\) in \(\mathbb C\). The classic Bernstein-Walsh theorem describes the relation between \(E_n(f,K)\) and the holomorphy of \(f\) on the set bounded by the level line of the Green function \(g_K(z)\). Some results are proved analogous to this theorem using compacts \(K\) of zero logarithmic capacity. Let \(K'\) be the set of limit points of an arbitrary compact \(K\subset \mathbb C\). If a function \(f(z)\) is defined on \(K\) then \(|f|_K=\sup\{|f(z)|: z\in K\}\). Denote by \(\mathfrak M_K\) the set of weak limit points of a sequence of measures and define for every \(z_0\in \mathbb C\setminus K'\) the functions \(U_H(z_0)=\inf_{\nu\in \mathfrak M_K} U^{\nu}(z_0)\), \(U_B(z_0)=\sup_{\nu\in \mathfrak M_K} U^{\nu}(z_0)\) satisfying for every \(b\in K'\) the following conditions: \[ (1)\quad \lim_{\substack{ z\to b\\ z\in \mathbb C\setminus K' }} U_H(z)=+\infty,\qquad\quad (2)\quad \lim_{\substack{ z\to b\\ z\in \mathbb C\setminus K' }} U_B(z)=+\infty. \] Now let \(f\) be a function given on a compact \(K\) in \(\mathbb C\) with \(\ln \text{cap}K=0\). If condition (2) is satisfied and for some \(R> 0\) the function \(f\) belongs to a determined algebra of holomorphic functions, then \[ {\varlimsup_{n\to\infty}}(E_n (f,K)\bigm/ |T_{n+1} |_K)^{1/n}=R^{-1}<\infty, \tag{3} \] where \(T_n\) is a Chebyshev polynomial of order \(n\). Conversely, if (1) and (3) are satisfied, then \(f\) belongs to an other (determined) algebra of holomorphic functions. In particular, \[ (f\in \text{Hol} K) \iff ({\varlimsup_{n\to \infty}} (E_n(f,K)\bigm/|T_{n+1}|_K)^{1/n}<\infty), \] where \(\text{Hol}K\) is the algebra of holomorphic functions on \(K\).