A note on the optimal recovery of functions in H∞

AbstractThe error of the best approximation of functions ƒ ϵ H∞ on the basis of given Hermitian data {ƒ(λ)(xk), k = 1, …, n, λ = 0, …, vk − 1} is expressed by the Blaschke product B(x̄; t) with zeros x̄ = (x1,…, xn) of multiplicities v1, …, vn, respectively. Given (vk)1n, we prove the uniqueness of the nodes x̄∗ which are optimal of type (v1, …, vn), i.e., which minimize the uniform norm of B(x̄; ·) in [a, b] ƒ (−1, 1) over a ⩽ x1 ⩽ … ⩽ xn ⩽ b. The extremal function B(x̄∗; t) is characterized by an oscillation property. Finally, a comparison theorem is proved, showing the dependence of the error on the order of the derivatives used in the information data