Density of Sampling and Interpolation in Reproducing Kernel Hilbert Spaces

Abstract

We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off-diagonal decay or even some weaker form of localization, then there exists a critical density $D$ with the following property: a set of sampling has density $\ge D$, whereas a set of interpolation has density $\le D$. The main theorem unifies many known density theorems in signal processing, complex analysis, and harmonic analysis. For the special case of bandlimited function we recover Landau's fundamental density result. In complex analysis we rederive a critical density for generalized Fock spaces. In harmonic analysis we obtain the first general result about the density of coherent frames.

28 pages

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• Density results for continuous frames
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• Smooth lattice orbits of nilpotent groups and strict comparison of projections
• Bernstein Spaces on Siegel CR Manifolds
• Sampling in the shift-invariant space generated by the bivariate Gaussian function
• Density of Gabor Systems Via the Short Time Fourier Transform
• Carleson and Sampling measures on Bernstein spaces on Siegel CR Manifolds
• Random periodic sampling patterns for shift-invariant spaces
• Necessary Density Conditions for Sampling and Interpolation in Spectral Subspaces of Elliptic Differential Operators
• Demystifying Carleson Frames