Blaschke products and optimal recovery in H ^∞

We examine the error in the optimal estimation of ∫ -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. Here f is known to be an analytic function in the unit disc which is bounded by l and w is a given weight function with prescribed behavior near ±1. A major role in our investigations is played by bounds on (-1, 1) from above and below for the finite Blaschke product which is based in the nodes of the quadrature formula. Optimal estimation of the function f, rather than its integral, is also studied. © 1987 Instituto di Elaborazione della Informazione del CNR.