On uniform approximations by polyanalytic polynomials on compact subsets of the plane (Q2769198)

cet \(D\subset \mathbb C\) be a simple connected domain bounded by a Jordan curve \(\Gamma\). Let \(\varphi :\overline D \to \overline {\Delta}:=\{|w|\leq 1\}\) be a homeomorphism such that \(\varphi\) is holomorphic in \(D\). Let \(m\) be the Lebesgue measure on the unit circle \(\{|w|=1\}\). Define a measure \(m_{\Gamma}\) on \(\Gamma\) by setting \(m_{\Gamma}(S):=m(\varphi (S))\) for Borel subsets \(S\) of \(\Gamma\). Given an integer \(n\geq 2\), let \(\mathcal P_n(\Gamma)\) be the closure in \(\mathcal C(\Gamma)\) of all complex valued polynomials \(p\) (of two real variables \(x_1, x_2\)) satisfying the equation \((\frac {\partial}{\partial \overline z})^n p =0\) on \(\mathbb C\), where \(z=x_1+ix_2\). NEWLINENEWLINENEWLINEMain result: \(\mathcal P_n(\Gamma) \neq \mathcal C(\Gamma)\) if and only if there exist two bounded holomorphic functions \(f, g \) \((g \not\equiv 0)\) on \(D\) such that \(\overline {\zeta} = \frac {f(\varphi (\zeta))}{g(\varphi (\zeta))}\) a.e. on \(\Gamma\) with respect to the measure \(m_{\Gamma}\) (the equality is understood in the sense of angular limit values). In his earlier paper [Math. Notes 59, 435-439 (1996; Zbl 0879.30021)] the author obtained the same result for rectifiable \(\Gamma\).