Polynomial approximation on compact sets bounded by Dini-smooth arcs
In this paper, the authors use the Faber operator to derive several results on the error of uniform polynomial approximation on a compact subset \(K\) of the complex plane \textbf{C}. They consider the Banach space \(A(K)\) of functions that are continuous on \(K\) and holomorphic in the interior of \(K\) endowed with the maximum norm \(\| \cdot \| _K\), and study the quantity \(E_n(F,K) := \inf \{\| F - P\| _K: \, P\) is a polynomial of degree \(\leq n\}\) \((n \in\) \textbf{N} := \(\{1,2,...\})\). It is assumed that \(K\) is not a single point and that its complement in the extended plane \textbf{C}\(_\infty\) is a simply connected region, so that there exists a unique biholomorphic mapping \(\psi :\)\textbf{C}\(_\infty \setminus \overline{D} \to\) \textbf{C}\(_\infty \setminus K\) (\(D :=\) the open unit disk) with \(\psi (\infty ) = \infty\) and \(\psi '(\infty ) > 0\) and the Faber polynomials \(F_n\) with respect to \(K\) are well defined. Then \(K\) is called a Faber set if the mapping \(T:(P,\| \cdot \| _D) \to (P,\| \cdot \| _K)\) (\(P := \{\)the polynomials\(\}\)) given by \((Tp)(z) = \sum_{\nu = 0}^n a_\nu F_\nu (z)\) (if \(p(z) = \sum_{\nu = 0}^n a_\nu z^\nu\)) is continuous. In such a case \(T\) extends to a one-to-one continuous linear operator \(T:A(\overline{D}) \to A(\overline{K})\). By using the boundedness of \(T\), a large number of results concerning polynomial approximation had been made for the case that \(K\) is the closure of a Jordan domain. In the current paper, it is shown that the corresponding methods also work for compact sets \(K\) which are not the closure of Jordan domains. The following theorem is selected among the diverse statements proved in the paper: Let \(K\) be a Faber set such that \(\psi\) is \(\beta\)-lipschitzian on the boundary \(\partial D\) for some \(\beta > 0\), and let \(\alpha \leq 1\). Then there exists a constant \(c = c(K, \alpha \beta )\) such that \(E_n(F,K) \leq cM/n^{\alpha \beta}\) for all \(n \in\) \textbf{N} and all \(F \in A(K)\) which is \(\alpha\)-lipschitzian on \(\partial K\) with constant \(M\). This is a Jackson-type result that can be applied to compact sets having Dini-smooth boundary. In addition, in the case of compact sets consisting of the closure of two Jordan domains touching at one point, an approximation result concerning those functions \(F\) such that a number of its derivatives extend continuously to \(K\) is derived. Moreover, it is obtained that some Bernstein-type results --that is, results on the sharpness of the rate of polynomial approximation -- are valid for this kind of domains. Finally, the authors prove that if \(K\) is itself a Jordan arc \(\Gamma\) with a special shape and \(\psi\) is \(1\)-lipschitzian on \(\partial D\) then \(\Gamma\) has the Jackson property, that is, there exists a constant \(d\) with \(E_n(F,\Gamma ) \leq d \cdot \omega (F,1/n)\) (\(n \in\) \textbf{N}, \(F\) continuous on \(\Gamma\)), where \(\omega (F, \cdot )\) is the modulus of continuity of \(F\) on \(\Gamma\).
scientific article; zbMATH DE number 2112938
scientific article