Blaschke products and optimal recovery in \(H^{\infty}\)
The error in the optimal estimation of \(\int^{1}_{-1}f(t)w(t)dt\) by a quadrature formula using values of f at equally spaced points of (-1,1) or at the zeros of ultraspherical polynomials is studied. Here f is a holomorphic function on the unit disk which is bounded by 1 and w is a weight function with prescribed behaviour near \(\pm 1\). The study is based on a detailed investigation of bounds on (-1,1) for the finite Blaschke product formed by the nodes of the quadrature formula. Optimal estimation of the function f, rather than its integral, is also investigated.
scientific article; zbMATH DE number 4044176