Universal Faber series with Hadamard-Ostrowski gaps
Let \(\Gamma\) be a nondegenerated continuum in the complex plane having connected complement. Then the Faber polynomials \(F_n (\Gamma)\) with respect to \(\Gamma\) are defined in terms of the suitably normalized Riemann mapping of \(|w| > 1\) onto the complement of \(\Gamma\). Moreover, each \(f\) holomorphic in a neighborhood of \(\Gamma\) has a Faber expansion of the form \(\sum\limits_{n=0}^\infty a_n (f,\Gamma) F_n (\Gamma)\) with Faber coefficients \(a_n (f,\Gamma)\). Suppose \(\Omega\) to be a domain in the complex plane with compact complement \(K\) and so that each component of \(K\) touches the same maximal level curve \(C(\Gamma,R^\ast)\) of the Riemann mapping. If \(H(\Omega)\) denotes the space of functions holomorphic in \(\Omega\) endowed with the topology of locally uniform convergence, the authors prove that the set of all \(f\in H(\Omega)\) with the property that the partial sums of the Faber expansion are uniformly universal on \(K\) and compactly convergent to \(f\) on \(B(\Gamma,R^\ast)\setminus K\), where \(B(\Gamma,R^\ast)\) is the union of the level curve \(C(\Gamma,R^\ast)\) and its interior, is residual in \(H(\Omega)\). Moreover, they prove that for all functions \(f\) with this property the Faber expansion has Hadamard-Ostrowski gaps.
scientific article; zbMATH DE number 5723305
scientific article