Boundary value problems and best approximation by holomorphic functions (Q1813784)

A classical problem in complex analysis consists in finding the distance of a function \(f\in L^ \infty\) on the unit circle \(T\) to \(H^ \infty\), the space of functions which extend to a bounded holomorphic function in the unit disk \({\mathcal D}\). The author studies the following generalized best approximation problem: Given a function \(F: T\times\mathbb{C}^ N\to R\) and distance \(d_ F(f,w):=\hbox {ess sup}_{t\in T} F(t,w(t))\), find \(w_ 0\in H^ \infty\) satisfying \(d_ F(f,w_ 0)=d_ F(f,H^ \infty):=\inf_{w\in H^ \infty}d_ F(f,w)\). The existence, uniqueness and the characterization of a generalized best approximation \(w_ 0\) under some quite general assumptions on the distance function \(F\) are given. New approach for determining \(w_ 0\) is based on the author's results about parameter-depending boundary value problems [Complex Variables, Theory Appl. 11, No. 3/4, 233-256 (1989; Zbl 0638.30040)]. Helton's conjecture [\textit{J. W. Helton}, J. Oper. Theory 15, 359-375 (1986; Zbl 0605.46043)] is also proved.