Invariances of frame sequences under perturbations

Duals and invariances of frame sequences

“…30 The equivalence of these parts with (b), (c), and (d) is proved by the authors. 8 We write U ∼ = V to mean that U is isomorphic to V , which is the case if and only if U and V have the same dimension. In the statement of this theorem the symbols P U | V denote the restriction of the orthogonal projection P U to the domain V. (a) 0 < R(U, V ) and 0 < R(V, U ).…”

“…8 Moreover, these types of perturbations preserve many of the fundamental properties of frames, including rank, excess, and deficit. Thus, Paley-Wiener-type perturbations preserve the "size" of a frame sequence in many ways.…”

“…The second group of results, due to Bishop, Heil, Koo, and Lim, 8 determines the exact relationships that hold among the major Paley-Wiener perturbation theorems for frame sequences, and addresses the invariance properties of frame sequences under such perturbations. Major properties of a frame sequence such as excess, deficit, and rank are shown to remain invariant under Paley-Wiener perturbations, but need not be preserved by compact perturbations.…”

“…The statements of some of these results are announced here, with full details and proofs in a separate research article. 8 …”

Duals of Frame Sequences

“…The equivalence of parts (a), (e), and (f) is stated in [24]. The equivalence of these parts with (b), (c), and (d) is proved in [3]. We write U ∼ = V to mean that U is isomorphic to V , which is the case if and only if U and V have the same dimension.…”