Uniform \(n\)-analytic polynomial approximations of functions on rectifiable contours in \(\mathbb{C}\)
A complex valued function \(f\) is said to be \(n\)-analytic in the complex domain \(D\) if \(\partial^nf/\partial \overline{z}^n =0\) on \(D\). An \(n\)-analytic polynomial is a polynomial in \(z\) and \(\overline{z}\) that is an \(n\)-analytic function in the whole plane. Let \(X\) be a closed rectifiable Jordan curve in \(\mathbb{C}\) and denote by \(C(X)\) the usual Banach space of continuous complex valued functions on \(X\) with the uniform norm. Using analytic properties, the author characterizes such curves \(X\) for which there exists at least a function in \(C(X)\) that cannot be limit of any sequence of \(n\)-analytic polynomials.
scientific article; zbMATH DE number 1028545