Joint approximation in BMOA and VMOA (Q1306861)

On the unit disc \({\mathbb{D}}\), consider some space \(A\) of analytic functions endowed with a topology \(\tau\). A relatively closed subset \(X\) of \({\mathbb{D}}\) is called a Mergelyan (resp., Farrell) set for \((A, \tau)\) if for every \(f\in A\) such that the restriction \(f|_X\) is uniformly continuous (resp., bounded) there is a sequence \(\{ p_n \}\) of analytic polynomials such that \(p_n\to f\) in \(\tau\) and \(p_n\to f\) uniformly on \(X\) (resp., pointwise on \(X\), and \(\sup_{z\in X}|p_n(z)|\to\sup_{z\in X}|f(x)|\)). Next, \(X\) is said to satisfy the nontangential condition if for almost every \(\zeta\in{\overline X} \cap \partial\mathbb{D}\) relative to Lebesgue measure on \(\partial\mathbb{D}\) there exists a sequence \(\{\zeta_n\}\subset X\) converging to \(\zeta\) nontangentially. It is shown that the nontangential condition is sufficient for \(X\) to be a Mergelyan set, and for \(X\) to be a Farrell set for the norm topology of \(\text{ VMOA}\); it becomes necessary and sufficient (in both cases) if we pass to the \(\text{ weak}^*\) topology of \(\text{ BMOA}\).