Positive harmonic functions on Denjoy domains in the complex plane

On Approximation of Continuous Functions by Entire Functions on Subsets of the Real Line

“…Proof. We essentially follow the outline of the proof of [6,Lemma 4]. Let U ± j be the open interval with the center in (c ± j + d ± j )/2 and the length…”

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

“…Therefore, by the Ahlfors criterion (see [13, p. 100]), L is quasiconformal. Thus, we can use the known techniques for approximation of continuous functions on quasiconformal arcs (see [5,Chapter 5] or [4] for more details). Let Φ : C \ L → D * := C \ D(1) be a conformal mapping normalized at infinity by the conditions…”

Lipschitz continuity of the Green function in Denjoy domains

“…(Added in August 2007.) V. V. Andrievskii, in the preprint [2], has found a different capacitary/metric covering condition on L\E that characterizes the case where dim P E =2 when n=2, and has also related this condition to the boundary behaviour of the Green function. His methods are based on conformal mappings and estimation of the modules of families of curves.…”