Minimal Interpolation by Blaschke Products II

Blaschke products and Nevanlinna-Pick interpolation

“…A similar conclusion was obtained in [4] in the rather special situation where S = {z n } is an interpolating sequence for the bounded analytic functions in D and where lim sup |w n | is sufficiently small. The present work complements the results in [4] and [5] as follows. First we find a rather precise condition on the sequence S = {z n } giving that for any indeterminate problem ( * ) all I α are Blaschke products.…”

“…For this reason the set {I α , 0 ≤ α < 2π} are often called the extremal solutions of ( * ). In [5] it was proved that for almost all α, I α is a Blaschke product, where the exceptional set of α-values has zero logarithmic capacity. If the interpolation problem ( * ) involves only a finite number of points, it is a classical fact that all I α are Blaschke products.…”

Interpolating sequences and the Nevanlinna Pick problem

“…This work is a continuation of [9] and [10] . We use the book [2] by J. Garnett as a referente for the theory of the classical Hardyspaces HP, 0 < p < co in D.…”

Boundary interpolation and approximation by infinite Blaschke products

“…The proof is similar to the one in [14, p. 496] that was used to obtain functions that interpolate sequences in the maximal ideal space of H ∞ , as opposed to sequence of points on the boundary of the disc. (See also [28].) Corollary 3.6.…”

“…Then every function in S is a solution of this problem and therefore there are (at least) two distinct solutions to this interpolation problem. By Stray's theorem [28], there is a Blaschke solution b. Now b − I = Bk for some k ∈ H ∞ and B(rz j ) → 0 as r → 1.…”