The Homogeneous Approximation Property and the Comparison Theorem for Coherent Frames

We show that the homogeneous approximation property and the comparison theorem hold for arbitrary coherent frames. This observation answers some questions about the density of frames that are not covered by the theory of Balan, Casazza, Heil, and Landau. The proofs are a variation of the method developed by Ramanathan and Steger. Frames provide redundant non-orthogonal expansions in Hilbert space, intuitively they should therefore be “denser” than orthonormal bases. The first fundamental result is Landau’s necessary condition for sets of sampling of band-limited functions [11]. A second fundamental idea is the homogeneous approximation property and the comparison theorem of Ramanathan and Steger [12] in the context of Gabor frames. Since then many contributions have investigated the density of frames and varied and applied the method of Ramanathan and Steger. See [3,4,8,13] for a sample of papers. The approach in [12] culminates in the deep investigation of Balan, Casazza, Heil, and Landau [1, 2] who have found a density theory for a general class of frames that are labeled by a discrete Abelian group. However, the BCHL theory is not universally applicable, for instance, the density of wavelet frames does not fall under this general theory, and a separate investigation is necessary, as was shown in a sequence of papers by Heil and Kutyniok [9,10] and by Sun [14–16]. Their work uses quite explicit and long calculations in the group of affine transformations. In this note we show that the approach of Ramanathan and Steger yields the homogeneous approximation property and a comparison theorem for the class of coherent frames. These frames arise as subsets of the orbit of a square-integrable group representation. In particular, wavelet frames possess this structure. The advantage of the abstract point of view is the conciseness and simplicity of the proofs and the much more general range of validity. The additional insight is that the homogeneous approximation property and the comparison theorem are a consequence of structure alone, we do not need to assume additional localization properties as in [1]. We will work in the context of a locally compact group G with Haar measure dx. We write |U | = ∫ G χU(x) dx for the measure of a set U ⊆ G and K c for the complement of K in G.