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 $\bar{\partial }$) 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.

arXiv comment33 pages

References in corpus (5)

• Deformation of Gabor systems
• Density of Sampling and Interpolation in Reproducing Kernel Hilbert Spaces
• Equidistribution estimates for Fekete points on complex manifolds
• Stability of Gabor frames under small time Hamiltonian evolutions
• Interpolating and sampling sequences for entire functions

Cited by in corpus (7)

• Density results for continuous frames
• Sampling in spaces of entire functions of exponential type in ${\mathbb{C}}^{n+1}$
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• Demystifying Carleson Frames
• Necessary Density Conditions for Sampling and Interpolation in Spectral Subspaces of Elliptic Differential Operators
• Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants
• Non-uniformly flat affine algebraic hypersurfaces