Optimal recovery of derivatives of bounded analytic and harmonic functions from inaccurate data (Q1326896)

The problem of finding the supremum \(\sup | f' (x) |\), \(| x | < 1\) among all analytic (and harmonic) functions in the unit disk \(\mathbb{D}\) bounded by 1 there, and satisfying additional growth restrictions on the interval \((-1,1)\) is solved explicitly. The solutions are given in terms of the infinite Blaschke product, solving the Milloux extremal problem [\textit{M. Heins}, Am. J. Math. 67, 212-234 (1945; Zbl 0060.217)]. The solution is based on a general functional-analytic duality approach to this and similar problems as, e.g., in \textit{S. Ya. Khavinson}, Transl., II. Ser., Am. Math. Soc. 129, 63-114 (1986; Zbl 0585.30003).