### Abstract

We are concerned with interpolation problems in H∞ where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence {zj} in the unit disk, we ask whether there exists a nontrivial minorant {ej} (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem f(zj) = aj has a nonvanishing solution f ∈ H∞ whenever 1 ≥ |aj| ≥ ej for all j. The sequences {zj} 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∞ ∩ VMO. © 2007 American Mathematical Society.

Original language | English |
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Pages (from-to) | 4449-4465 |

Journal | Transactions of the American Mathematical Society |

Volume | 359 |

Issue number | 9 |

DOIs | |

Publication status | Published - 1 Dec 2007 |

### Keywords

- Nonvanishing analytic functions
- Thin interpolating sequences

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## Cite this

Dyakonov, K., & Nicolau, A. (2007). Free interpolation by nonvanishing analytic functions.

*Transactions of the American Mathematical Society*,*359*(9), 4449-4465. https://doi.org/10.1090/S0002-9947-07-04186-4