A note on the optimal recovery of functions in H∞

Abstract

The error of the best approximation of functions ƒ ϵ H ∞ on the basis of given Hermitian data {ƒ (λ)(x k), k = 1, …, n, λ = 0, …, v k − 1} is expressed by the Blaschke product B( x ̄ ; t) with zeros x ̄ = (x 1,…, x n) of multiplicities v 1, …, v n , respectively. Given ( v k ) 1 n , we prove the uniqueness of the nodes x ̄ ∗ which are optimal of type ( v 1, …, v n ), i.e., which minimize the uniform norm of B( x ̄ ; ·) in [a, b] ƒ (−1, 1) over a ⩽ x 1 ⩽ … ⩽ x n ⩽ b. The extremal function B( x ̄ ∗; 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.