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Free interpolation by nonvanishing analytic functions. (English) Zbl 1141.30010

Carleson’s celebrated interpolation theorem states that the interpolating sequences (for bounded analytic functions) in the unit disk are precisely the ones uniformly separated in the (pseudo-) hyperbolic metric:
\[ 0<\inf_{j} \prod_{k\colon k\neq j} | {z_j-z_k \over 1-\bar{z_j} z_k}| \,. \]
As the title of the present paper indicates, the authors are concerned with a related problem of free interpolation by non-vanishing (and bounded) analytic functions in the unit disk, which they solve completely. To phrase the questions precisely, a sequence of positive numbers \((\varepsilon_j)_j\) is said to be a minorant if it is bounded by one and tends to zero. The non-vanishing interpolation problem is: describe all sequences \((z_j)_j\) in the disk for which there exists a nontrivial minorant \((\varepsilon_j)_j\) such that for every sequence \((a_j)\) with \(\varepsilon_j\leq | a_j| \leq 1\) there exists a non-vanishing bounded analytic solution \(f\) such that \(f(z_j)=a_j\) for all \(j\). It should be intuitively clear (but far from obvious) that “only a few” interpolating sequences suffice for this purpose, namely the ones which are “fairly sparse” in some sense. The authors prove that this is indeed the case: non-vanishing interpolation is possible if and only if the sequence \((z_j)_j\) is thin, that is,
\[ \lim_{j\to\infty} \prod_{k\colon k\neq j} | {z_j-z_k \over 1-\bar{z_j} z_k}| = 1 \,. \]
Such sequences have been appearing frequently in function theory, ever since the thesis of the late T. Wolff at Berkeley in 1979. This nice paper contains other results too, and ends with several related open questions (e.g., for non-vanishing interpolation by outer functions).

MSC:

30D50 Blaschke products, etc. (MSC2000)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30H05 Spaces of bounded analytic functions of one complex variable
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References:

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