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 $\ell_A^p$ of functions analytic on the unit disk for every $1\le p<2$. For $1<p<2$, a single Blaschke set $W$ is a zero set for every $\ell_A^p$ in this range, but is not exact for any of them. The endpoint case $p=1$, corresponding to the analytic Wiener algebra $\ell_A^1$, 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 $\ell^p$ with pairwise trivial intersections, recovering a known existence result.