Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

Abstract

Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density.The techniques combine methods from several complex variables (estimates for∂) and the theory of localized frames in general reproducing kernel Hilbert spaces (with no analyticity assumed). The abstract results on Fekete points and deformation of frames may be of independent interest.

Sampling in spaces of entire functions of exponential type in Cn+1

“…In [GL20] the authors obtain very interesting results concerning some necessary and some sufficient condition for sampling in Fock spaces in C 2 in connection with existence of Gabor frames in R 2 . In [GHOCR19] the authors prove strict density inequalities for sampling and interpolation in Fock spaces in C d defined by a plurisubharmonic weight. See also [GJM20] for related results, and the references in the cited papers.…”

Nonuniformly Flat Affine Algebraic Hypersurfaces

“…Nevertheless, the density does have to be somewhat constrained. An interesting necessary condition was introduced in [L-1997], and very recently improved in [GHOR-2018].…”

Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of $L^p({\mathbb R}^n)$

“…The space V 2 φ is a localizable reproducing kernel Hilbert space in L 2 (R 2n ), see [18]. If β ∈ (0, 2/n), then Γ = βZ 2n is a relatively separated set and there exists {F γ : γ ∈ Γ} frame for V 2 φ with frame bounds A = 1 4β 2n and B = 3 2 .…”