On the dependence of uniform polyanalytic polynomial approximations on the order of polyanalyticity (Q948554)

Let \(U\) be an open subset of the complex plane \(\mathbb{C}.\) A function \(f\) is said to be n-analytic in \(U\) if it is of the form \(f(z)=\overline z^{n-1} f_{n-1}(z) + \dots + \overline z f_1(z) + f_0(z),\) where \(n\in \mathbb{N}\) and \(f_0,f_1,\dots , f_{n-1}\) are holomorphic functions in \(U.\) For a compact subset \(X\) in \(\mathbb{C}\) let \(P_n(X)\) denote the closure in \(\mathcal{C}(X)\) of the set of all n-analytic polynomials. It is shown that for each integer \(n\geq 1\) there exists a compact set \(X\subset \mathbb{C}\) such that \(P_{2n}(X)=\mathcal{C}(X)\neq P_n(X).\)