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Free interpolation by nonvanishing analytic functions
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We are concerned with interpolation problems in $ H^\infty$ where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence $ \{z_j\}$ in the unit disk, we ask whether there exists a nontrivial minorant $ \{\varepsilon_j\}$ (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem $ f(z_j)=a_j$ has a nonvanishing solution $ f\in H^\infty$ whenever $ 1\ge\vert a_j\vert\ge\varepsilon_j$ for all $ j$. The sequences $ \{z_j\}$ with this property are completely characterized. Namely, we identify them as 'thin' sequences, a class that arose earlier in Wolff's work on free interpolation in $ H^\infty\cap$ VMO.
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DYAKONOV, Konstantin M. and NICOLAU, Artur. Free interpolation by nonvanishing analytic functions. Transactions of the American Mathematical Society. 2007. Vol. 359, num. 9, pags. 4449-4465. ISSN 0002-9947. [consulted: 15 of August of 2026]. Available at: https://hdl.handle.net/2445/97263