Duality in reconstruction systems

Abstract

We consider the notion of finite dimensional reconstructions systems (RS's), which includes the fusion frames as projective RS's. We study erasures, some geometrical properties of these spaces, the spectral picture of the set of all dual systems of a fixed RS, the spectral picture of the set of RS operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting. WeIn the case J = {j}, the lower bound in Theorem 3.3 is greater than that obtained in [3]:Proposition 3.4. Let V, and M J be as in Theorem 3.3, with J = {j}. ThenProof. We can suppose M −1 J ≥ 1, since otherwise (7) is evident. Note that1. The vector µ ∈ Λ(D(V)).1. The set Λ(OP v ) is convex.

Harmonic reconstruction systems

“…Harmonic Reconstruction Systems 693 duced in [12] and m, n, F d -RS's are considered in [13]. RS's are a generalization of frames [3,6,10] and fusion frames (or frames of subspaces) [4,5].…”

“…Let A ∈ F d×m and {D g A} g∈G ∈ RS G, m, F d where {D g } g∈G is obtained as in Theorem 2. 13. ρ : G → U F d given by ρ(g) = D g is a reducible representation of G that can be decomposed as a sum of one-dimensional irreducible…”

“…Example 3. 13. By Corollary 3.12, if in Example 3.8 and Example 3.9 n is prime and we only consider the restriction onto the component i, we obtain a description of sets I ⊆ {1, .…”

Group reconstruction systems

“…If n 1 = · · · = n m = n, we write (m, n, H)-RS. The concept of (m, n, H)-RS (with F n replaced by any Hilbert space K of dimension n) was introduced in [15] and m, n, F d -RS's are considered in [16]. In [20], RS's for non necessarily finite dimensional Hilbert spaces are called g-frames and are shown to be equivalent to stable space splittings of Hilbert spaces [18].…”

“…They are collections of weights and orthogonal projections, and permit us to recover an element f ∈ H from packets of linear coefficients. For finite dimensional Hilbert spaces H, reconstruction systems (RS's) [15,16], that are collections of operators that provide a The relation between the irreducibility of the representation and the tightness of the GRS is considered in Section 6. We prove that the GRS generated by an irreducible representation is tight (Theorem 6.2, Theorem 6.3 and Proposition 6.4).…”