Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
Abstract
We establish sharp sampling and interpolation criteria for the radial--angular Gaussian mixed-norm Fock spaces ℱαp,q, where α>0 and 0<p,q≤∞. The natural point-evaluation scale leads to the normalized restriction Rα,Λp,qf = (f(λ)e−α/2|λ|2(1+|λ|)1/q−1/p)λ∈Λ. With this normalization, the critical density is the classical value α/π throughout the Banach and quasi-Banach ranges. For an arbitrary locally finite set Λ, sampling is equivalent to Dsep−(Λ)>α/π, together with relative separation when p<∞; no relative-separation condition is required when p=∞. Here Dsep− is the supremum of the lower Beurling densities over separated subsets of Λ. Interpolation is equivalent to separation and D+(Λ)<α/π. The same criterion characterizes interpolation for the little endpoint fαp,∞, and in both settings the normalized restriction map admits a bounded linear right inverse. The sufficiency arguments are based on rapidly localized Hilbert dual and Lagrange atoms together with a weighted localized synthesis theorem on the full mixed-norm scale. For necessity, we prove that lower stability of a rapidly localized matrix on a weighted annular mixed sequence space implies lower stability on ℓ2. The proof first passes to ℓ∞ by translated polynomial cutoffs and a commutator estimate, and then uses the p-independence theorem for the Sjöstrand class. Applied through a fixed Hilbert sampling lattice, this reduces the strict density conditions to the classical Hilbert Fock sampling and interpolation theorems.
Related
- Positive Toeplitz Operators Between Gaussian Mixed-Norm Fock Spaces
- Relevant sampling of non-decaying signals in Orlicz and mixed-norm Orlicz spaces defined on a locally compact group
- Sharp Gaussian Mixed-Norm Theory for Fock Spaces
- Fock Projections between Gaussian Mixed-Norm Spaces: Full-Range Boundedness and Compactness
- Minimum-Norm Interpolation in Weighted Variation Spaces: Direct and Inverse Problems
BibTeX
@misc{fang2026sampling,
title = {{Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces}},
author = {Xiang Fang and Pham Trong Tien},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23572},
eprint = {2609.23572},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23572},
}
RIS
TY - UNPB
TI - Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
AU - Xiang Fang
AU - Pham Trong Tien
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23572
UR - https://arxiv.org/abs/2609.23572
AB - We establish sharp sampling and interpolation criteria for the radial--angular Gaussian mixed-norm Fock spaces ℱ_α^p,q, where α>0 and 0<p,q≤∞. The natural point-evaluation scale leads to the normalized restriction R_α,Λ^p,qf = (f(λ)e^−α/2|λ|^2(1+|λ|)^1/q−1/p)_λ∈Λ. With this normalization, the critical density is the classical value α/π throughout the Banach and quasi-Banach ranges. For an arbitrary locally finite set Λ, sampling is equivalent to D_sep^−(Λ)>α/π, together with relative separation when p<∞; no relative-separation condition is required when p=∞. Here D_sep^− is the supremum of the lower Beurling densities over separated subsets of Λ. Interpolation is equivalent to separation and D^+(Λ)<α/π. The same criterion characterizes interpolation for the little endpoint f_α^p,∞, and in both settings the normalized restriction map admits a bounded linear right inverse. The sufficiency arguments are based on rapidly localized Hilbert dual and Lagrange atoms together with a weighted localized synthesis theorem on the full mixed-norm scale. For necessity, we prove that lower stability of a rapidly localized matrix on a weighted annular mixed sequence space implies lower stability on ℓ^2. The proof first passes to ℓ^∞ by translated polynomial cutoffs and a commutator estimate, and then uses the p-independence theorem for the Sjöstrand class. Applied through a fixed Hilbert sampling lattice, this reduces the strict density conditions to the classical Hilbert Fock sampling and interpolation theorems.
ER -