Hardy approximation to \(L^p\) functions on subsets of the circle with \(1\leq p<\infty\) (Q1381493)

Let \(K\) be a subset of the unit circle \(\Pi\), let \(f\in L^p(K)\), and let \(h\in L^p (\Pi \setminus K)\). For \(p\in[1,\infty)\) the authors consider the problem of finding a best approximant to \(f\) in \(L^p(K)\) among the functions of the form \(g| K\), where \(g\in H^p\) and \(\| g-h \|_{L^p (\pi \setminus K)}\leq M\). It is shown that such an approximant always exists and is unique; moreover, \(\| h-g \|_{L^p (\pi\setminus K)} =M\) for the function \(g_0 \in H^p\) corresponding to this approximant, unless \(f=g_0\) on \(K\). (Surely, the most difficult case is \(p=1\).) If \(p=2\), a nearly explicit procedure for finding \(g_0\) is described.