Density of sampling and interpolation in reproducing kernel Hilbert spaces

Abstract

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 ≥ D, whereas a set of interpolation has density ≤ 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.

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

“…We will show that the homogeneity of φ may be removed and that the necessary conditions (4) and (5) are valid for all weights φ satisfying (1). This more general result follows from the abstract density theory [4,12], which is applicable due to the off-diagonal decay of the reproducing kernel. See Section 2.3 for the details.…”

“…Br(z) (i∂∂φ) n . It can be shown by combining results of [19], [6] and [12] that in 1 dimension we have (6) D + φ (Λ) = 1 π n n! D + φ (Λ), and similarly for the lower densities.…”

“…D + φ (Λ), and similarly for the lower densities. The relation (6) holds also in several variables if we assume in addition that φ is 2-homogeneous [17] -see also [12,Section 5.4]. In general, it remains an open problem to decide when both densities coincide.…”

Monte Carlo wavelets: a randomized approach to frame discretization

“…Asymptotic regularization N − α 2α+2 6 Numerical implementation To implement the system of Monte Carlo wavelets in (11) we exploit equation (7), that is,…”

“…To construct a discrete frame it is common practice to first construct a frame on a continuous parameter (measure) space, where the mathematical properties are nice and the structure is rich, and then obtain a discrete frame by carefully selecting a discrete subset of parameters. The discretization problem has been studied extensively and in several settings [5,2,7,6]. Although theoretically relevant in establishing density theorems and complete characterizations, some general discretization procedures require assumptions which might be difficult to confirm, and they are often not constructive.…”