Minimal Interpolation by Blaschke Products

Abstract

STRAYCOROLLARY. If {z v } is an interpolating sequence and w v -*• 0 as v -• oo, then the interpolation problem (*) has a unique minimal solution / 0 = kB, being a complex multiple of a Blaschke product.Proofs of Theorem 1 and its corollary will be given in §2. The necessity of {z v } being an interpolating sequence is explained in §3.

Blaschke products and Nevanlinna-Pick interpolation

“…If the interpolation problem ( * ) involves only a finite number of points, it is a classical fact that all I α are Blaschke products. 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.…”

“…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.…”

Interpolating sequences and the Nevanlinna Pick problem

“…Our final result is only a slight extension of Theorem 1 in [9] . We include it here because of its relevante to recent work by T. Nazaki in [5] .…”

“…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.…”

Nevanlinna’s coefficients and Douglas algebras

“…Proof, (i), (ii), (iii) are well known (see [8] and the references there given to [2]). Using the relations in (i)…”