Jens Gerlach Christensen

Full length article: Sampling in reproducing kernel Banach spaces on Lie groups

We present sampling theorems for reproducing kernel Banach spaces on Lie groups. Recent approaches to this problem rely on integrability of the kernel and its local oscillations. In this paper, we replace these integrability conditions by requirements on the derivatives of the reproducing kernel and, in particular, oscillation estimates are found using derivatives of the reproducing kernel. This provides a convenient path to sampling results on reproducing kernel Banach spaces. Finally, these results are used to obtain frames and atomic decompositions for Banach spaces of distributions stemming from a cyclic representation. It is shown that this process is particularly easy, when the cyclic vector is a Garding vector for a square integrable representation.

Multi-window Gabor frames in amalgam spaces

We show that multi-window Gabor frames with windows in the Wiener algebra

Coorbit Description and Atomic Decomposition of Besov Spaces

W(L^{\infty}, \ell^{1})

SAMPLING IN SPACES OF BANDLIMITED FUNCTIONS ON COMMUTATIVE SPACES

are Banach frames for all Wiener amalgam spaces. As a byproduct of our results we positively answer an open question that was posed by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the continuity of the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra. The proofs are based on a recent version of Wieners

Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach

1/f

Atomic decompositions of mixed norm Bergman spaces on tube type domains

lemma.

Sampling, amenability and the Kunze-Stein phenomenon

Function spaces are central topics in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gröchenig [6–8] and then later extended in Christensen and Ólafsson [3] gives a unified method to construct Banach spaces of functions based on representations of Lie groups. In this article, we identify the homogeneous Besov spaces on stratified Lie groups introduced by Führ and Mayeli [12] as coorbit spaces in the sense of [3] and use this to derive atomic decompositions for the Besov spaces.

Examples of Coorbit Spaces for Dual Pairs

A homogeneous space \(\mathbf{X} = G/K\) is called commutative if G is a locally compact group, K is a compact subgroup, and the Banach ∗ -algebra \({L}^{1}{(\mathbf{X})}^{K}\) of K-invariant integrable functions on \(\mathbf{X}\) is commutative. In this chapter we introduce the space \({L}_{\Omega }^{2}(\mathbf{X})\) of \(\Omega \)-bandlimited function on \(\mathbf{X}\) by using the spectral decomposition of \({L}^{2}(\mathbf{X})\). We show that those spaces are reproducing kernel Hilbert spaces and determine the reproducing kernel. We then prove sampling results for those spaces using the smoothness of the elements in \({L}_{\Omega }^{2}(\mathbf{X})\). At the end we discuss the example of \({\mathbb{R}}^{d}\), the spheres S d , compact symmetric spaces, and the Heisenberg group realized as the commutative space \(\mathrm{U}(n) \ltimes {\mathbb{H}}_{n}/\mathrm{U}(n)\).

Coorbit spaces for dual pairs

Sampling theory is a fundamental area of study in harmonic analysis and signal and image processing. The purpose of this paper is to connect sampling theory with the geometry of the signal and its domain. It is relatively easy to demonstrate this connection in Euclidean spaces, but one quickly gets into open problems when the underlying space is not Euclidean. We focus primarily on Euclidean and hyperbolic geometries.There are numerous motivations for extending sampling to non-Euclidean geometries. Applications of sampling in non-Euclidean geometries are showing up areas from EIT to cosmology. Irregular sampling of bandlimited functions by iteration in hyperbolic space is possible, as shown by Feichtinger and Pesenson. Sampling in spherical geometry has been analyzed by many authors, e.g., Driscoll, Healy, Keiner, Kunis, McEwen, Potts, and Wiaux, and brings up questions about tiling the sphere. In Euclidean space, the minimal sampling rate for Paley-Wiener functions on \(\mathbb{R}^{d}\), the Nyquist rate, is a function of the bandwidth. No such rate has yet been determined for hyperbolic or spherical spaces. We look to develop a structure for the tiling of frequency spaces in both Euclidean and non-Euclidean domains. In particular, we establish Nyquist tiles and sampling groups in Euclidean geometry, and discuss the extension of these concepts to hyperbolic and spherical geometry and general orientable surfaces.

The Uncertainty Principle for Operators Determined by Lie Groups

We use the authors previous work on atomic decompositions of Besov spaces with spectrum on symmetric cones, to derive new atomic decompositions for Bergman spaces on tube type domains. It is related to work by Ricci and Taibleson who derived decompositions for classical Besov spaces from atomic decompositions of Bergman spaces on the upper half plane. Moreover, for this class of domains our method is an alternative to classical results by Coifman and Rochberg, and it works for a larger range of Bergman spaces.