Geometric aspects of frame representations of abelian groups

Abstract

Abstract. We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

Orthogonal frames of translates

“…Let G, H, and Θ H,G be as in Lemma 6. For singly generated systems, we recover the characterization developed in [4].…”

“…Proof. For singly generated systems, the Bessel condition is equivalent to the local integrability condition [4]. If the Bessel sequences are orthogonal, then for each…”

Frames and bases in tensor products of Hilbert spaces and Hilbert C*-modules

“…Let Ĝ denote the dual group of G, i.e., the group of characters on G and let λ be the normalized Haar measure on Ĝ. Let π : G −→ B(H) be a frame representation with frame vector v. As we have in [1,13,17] there is a spectral measure E on Ĝ such that…”

“…Since π is a frame representation, by using the results in §2 of [1] and the properties of spectral measure there is a unitary operator U: H −→ L 2 (F, λ |F), where F is a measurable subset of Ĝ with λ (F) > 0 and λ |F is the restriction of Haar measure λ to F such that U interwines the spectral measure on H and the canonical spectral measure on Ĝ. The operator U is called the decomposition operator.…”

Frame analysis and approximation in reproducing kernel Hilbert spaces

“…While frames already have impressive uses in signal processing (see, e.g., [ALTW04,Chr99]), they have recently [CCLV05,CKL04] been shown to be central in our understanding of a fundamental question in operator algebras, the Kadison-Singer conjecture. We refer the reader to [CFTW06] for up-to-date research, and to [Chr99,KaRi97,Nel57,Nel59] for background.…”