An asymptotically optimal algorithm for approximating bounded analytic functions (Q1358135)

This paper is devoted to construct a linear algorithm for approximating \(H^\infty\) functions whose modulus is bounded by one. Let \(E\) be a compact subset of the unit disk, such that \(\partial E\) belong to \(C^{2,\alpha}\). Let \[ B^*(z)= \prod^n_{k=1} {z-\alpha^*_k\over 1-\overline\alpha^*_kz}. \] The main result is the following estimate \[ |B^*|_{C(E)}\leq\exp\Biggl({- n\over\text{cap}(E, \Delta)}\Biggr)2\Biggl(1+O\Biggl({1\over n^\alpha}\Biggr)\Biggr), \] where \(C(E)\) is the space of continuous functions on \(E\), and \(\text{cap}(E,\Delta)\) is the Green capacity of \(E\) with respect to \(\Delta\). As a corollary it is obtained that \[ \sup_{f\in A}|f-P^*_nf|_{C(E)}\leq 2\Biggl(1+O\Biggl({1\over n^\alpha}\Biggr)\Biggr) \delta_n(A,C(E)), \] where \(P^*_n\) is the Fischer-Micchelli algorithm and \(\delta_n(A,C(E))\) is the linear \(n\)-widths of \(A\) in \(C(E)\). The above estimate means that the approximation is asymptotically optimal. The construction of \(P^*_n\) is based on sampling a function in the Fejér nodes on \(\partial E\).