Group reconstruction systems

Abstract

Abstract. We consider classes of reconstruction systems (RS's) for finite dimensional real or complex Hilbert spaces H, called group reconstruction systems (GRS's), that are associated with representations of finite groups G. These GRS's generalize frames with high degree of symmetry, such as harmonic or geometrically uniform ones. Their canonical dual and canonical Parseval are shown to be GRS's. We establish simple conditions for one-erasure robustness. Projective GRS's, that can be viewed as fusion frames, are also considered. We characterize the Gram matrix of a GRS in terms of block group matrices. Unitary equivalences and unitary symmetries of RS's are studied. The relation between the irreducibility of the representation and the tightness of the GRS is established. Taking into account these results, we consider the construction of Parseval, projective and one-erasure robust GRS's.

Harmonic reconstruction systems

“…It follows from Theorem 4.5 in [14] and Theorem 2.7. Suppose that T g = QD g A, for each g ∈ G, where Q, D g and A are as in the enunciation.…”

“…If m = 1, the ρ i are all different, therefore, by Theorem 6.10 in [14], {D g A} g∈G ∈ RS G, 1, 3. Harmonic reconstruction systems.…”

“…RS's are a generalization of frames [3,6,10] and fusion frames (or frames of subspaces) [4,5]. Concretely, an (m, 1, H)-RS is a frame and projective RS's can be viewed as fusion frames (see Remark 2.3 and Remark 2.4 in [14]). In [16] RS's for not necessarily finite dimensional Hilbert spaces are called g-frames and are shown to be equivalent to stable space splittings of Hilbert spaces [15].…”

“…In [14], two types of RS's associated with unitary representations of finite groups, called group reconstruction systems (GRS's) are studied. Here we consider the abelian case, i.e., GRS's associated with unitary representations of finite abelian groups.…”

“…For properties of RS's and GRS's and their relation with frames and fusion frames (or frames of subspaces), we refer the reader to [14].…”