Simultaneous approximation in function spaces (Q2758410)

Let \(A\) be a space of analytic functions on the unit disc \(D\), let \(X\) be a relatively closed subset of \(D\), and let \(B\) be some topological function space on \(X\). If \(B\) consists of all uniformly continuous functions on \(X\), then we call \(X\) a Mergelyan set for \(A\) if, for any \(f\in A\) with \(f|_X\in B\), there is a sequence \((p_n)\) of polynomials such that \(p_n\to f\) in \(A\) and \(p_n|_X\to f|_X\) in \(B\) (i.e. uniform convergence on \(X\)). In the corresponding situation where \(B\) denotes the space of all bounded continuous functions on \(X\) with the topology of bounded pointwise convergence, we call \(X\) a Farrell set for \(A\). The main problems under consideration in these lecture notes are to find geometric or topological characterizations of Farrell and Mergelyan ses. These questions are related to the simpler one of identifying the sets of determination for the class \(A\), that is, the sets \(X\subseteq D\) such that \(\sup_X|f|= \sup_D|f|\) for every \(f\in A\). The article surveys known characterizations of Farrell and Mergelyan sets for \(H^p(D)\) \((0< p\leq\infty)\), BMOA, \(H(D)\) (the space of all analytic functions on \(D\) with the topology of local uniform convergence), and the Dirichlet space. Some of these results are very recent. An attractive feature of the article is the collection of open problems that are posed. They should stimulate further resarch in this interesting field.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00045].