Orthogonal Frames of Translates

Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel sequences have the form of translates of a finite number of functions in L(R). The characterizations are applied to Bessel sequences which have an affine structure, and a quasi-affine structure. These also lead to characterizations of superframes. Moreover, we characterize perfect reconstruction, i.e. duality, of subspace frames for translation invariant (bandlimited) subspaces of L(R). Introduction Frames for (separable) Hilbert spaces were introduced by Duffin and Schaeffer [13] in their work on non-harmonic Fourier series. Later, Daubechies, Grossmann, and Meyer revived the study of frames in [12], and since then, frames have become the focus of active research, both in theory and in applications, such as signal processing. Every frame (or Bessel sequence) determines an analysis operator, the range of which is important for a number of applications. Information about this range is partially revealed by considering the composition of analysis and synthesis operators for different frames. We view this composition as a sum of rank one tensors. The present paper considers frames and Bessel sequences in L(R) which arise from translations of generating functions, such as in wavelet and Gabor frame theory. The goal is to determine when the infinite sum of rank one tensors involving these translations is actually the 0 operator. See the subsection entitled ”Motivation” below. 0.1. Definitions. LetH be a separable Hilbert space and J a countable index set. A sequence X := {xj}j∈J is a frame if there exist positive real numbers C1, C2 such that for all v ∈ H , (1) C1‖v‖ 2 ≤ ∑ j∈J |〈v, xj〉| 2 ≤ C2‖v‖ . If X satisfies only the second inequality, (i.e. only C1 = 0 satisfies the first inequality), then X is called a Bessel sequence. Given X which is Bessel, define the analysis operator ΘX : H → l (J) : v 7→ (〈v, xj〉)j; and the synthesis operator Θ∗X : l (J) → H : (cj)j 7→ ∑