The order of \(L^1\)-approximation by elements of the disc algebra (Q2054419)

\textit{D. Khavinson} et al. [Constr. Approx. 14, No. 3, 401--410 (1998; Zbl 0916.46040)] showed that, given a continuous function \(f\) on the unit circle \(\mathbb T\) with \(\|f\|_\infty=1\) and a function \(G\in H^1(\mathbb D)\) with \(\|f-G\|_1<\epsilon\), then there exists \(G^*\in A(\mathbb D)\) with \(\|G^*\|_\infty\leq 1\) and \[\|f-G^*\|_1\leq C\epsilon \log 1/\epsilon.\] In the paper under review it is proved that \(O(\epsilon \log 1/\epsilon)\) is the precise order. In fact, if \(f=f_\epsilon\) is given by \(f=F/|F|\) where \(F(z)=\exp(\epsilon^2/(1+\epsilon -z))\in A(\mathbb D)\), then \(\|f-F\|_1\leq C\epsilon^2\), but for any \(G^*\in A(\mathbb D)\) with \(\|G^*\|_\infty\leq 1\), one has \(\|f-G^*\|_1\geq c\; \epsilon^2\log 1/\epsilon\).