Zero sets in spaces of analytic functions with p-summable Taylor coefficients
Abstract
A zero set for a space of analytic functions is a set on which some nonzero function in the space vanishes; it is exact if it coincides with the full zero set of such a function. We construct zero sets that are not exact in the classical coefficient spaces ℓAp of functions analytic on the unit disk for every 1≤ p<2. For 1<p<2, a single Blaschke set W is a zero set for every ℓAp in this range, but is not exact for any of them. The endpoint case p=1, corresponding to the analytic Wiener algebra ℓA1, requires a separate construction, yielding a non-exact zero set that accumulates at only one point of the unit circle. The proofs combine quadratic characters, finite-dimensional Clark transforms, and logarithmic estimates in zero-free regions. We also establish sharp moment duality and an exact summability criterion for a family of regular polygons. For p>2, we construct two disjoint exact zero sets whose union is a uniqueness set. For these exponents, suitably chosen rotations also yield a continuum-sized family of nonzero closed shift-invariant subspaces of ℓp with pairwise trivial intersections, recovering a known existence result.
Related
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- Generalized tri-circular projections on some vector-valued spaces of analytic functions
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- On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves
- Meromorphic order from Taylor coefficients, least singular values of Hankel matrices and rational approximants
BibTeX
@misc{abakumov2026zero,
title = {{Zero sets in spaces of analytic functions with $p$-summable Taylor coefficients}},
author = {Evgeny Abakumov and Isabelle Chalendar and Alexei Poltoratski},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02159},
eprint = {2611.02159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02159},
}
RIS
TY - UNPB
TI - Zero sets in spaces of analytic functions with p-summable Taylor coefficients
AU - Evgeny Abakumov
AU - Isabelle Chalendar
AU - Alexei Poltoratski
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02159
UR - https://arxiv.org/abs/2611.02159
AB - A zero set for a space of analytic functions is a set on which some nonzero function in the space vanishes; it is exact if it coincides with the full zero set of such a function. We construct zero sets that are not exact in the classical coefficient spaces ℓ_A^p of functions analytic on the unit disk for every 1≤ p<2. For 1<p<2, a single Blaschke set W is a zero set for every ℓ_A^p in this range, but is not exact for any of them. The endpoint case p=1, corresponding to the analytic Wiener algebra ℓ_A^1, requires a separate construction, yielding a non-exact zero set that accumulates at only one point of the unit circle. The proofs combine quadratic characters, finite-dimensional Clark transforms, and logarithmic estimates in zero-free regions. We also establish sharp moment duality and an exact summability criterion for a family of regular polygons. For p>2, we construct two disjoint exact zero sets whose union is a uniqueness set. For these exponents, suitably chosen rotations also yield a continuum-sized family of nonzero closed shift-invariant subspaces of ℓ^p with pairwise trivial intersections, recovering a known existence result.
ER -