Duals of Frame Sequences

Abstract

Frames provide unconditional basis-like, but generally nonunique, representations of vectors in a Hilbert space H . The redundancy of frame expansions allows the flexibility of choosing different dual sequences to employ in frame representations. In particular, oblique duals, Type I duals, and Type II duals have been introduced in the literature because of the special properties that they possess. A Type I dual is a dual such that the range of its synthesis operator is contained in the range of the synthesis operator of the original frame sequence, and a Type II dual is a dual such that the range of its analysis operator is contained in the range of the analysis operator of the original frame sequence. This paper proves that all Type I and Type II duals are oblique duals, but not conversely, and characterizes the existence of oblique and Type II duals in terms of direct sum decompositions of H , as well as characterizing when the Type I, Type II, and oblique duals will be unique. These results are also applied to the case of shift-generated sequences that are frames for shift-invariant subspaces of L 2 (R d ).

The effect of perturbations of operator-valued frame sequences and fusion frames on their duals

“…In the case of vector sequences there exist several notions of duals of frame sequences (cf. [10,21] and also [17]). In what follows, we define duals of operator-valued frame sequences, thereby generalizing the notion of the so-called Type I duals of vector frame sequences from [21].…”

Applications of Parameterizations of Oblique Duals of a Frame Sequence

“…The original definition of oblique dual in [8] is different from the one we use, but the two definitions are equivalent [14,Theorem 1.4]. Note that the term oblique dual used in [3] is just a dual in our definition.…”

“…In this article, we first clarify the relationship among the various definitions of duals of a given frame sequence introduced and studied by Christensen and Eldar [8], Askari Hemmat and Gabardo [2,3], Andruchow, Antezana, and Corach [1], and Heil, Koo, and Lim [14]. In particular, if we fix the appropriate range spaces, there are two parameterizations of oblique duals of a given frame sequence in [8] and [1].…”

Duals of g-frames and g-frame sequences

“…Also, continuous g-frames are introduced in [1]. In this paper we generalize some results in [3] and [4] to g-frames and g-frame sequences.…”

“…We will construct a g-frame for V that is a Type II dual of {Λ i } i∈I . Recall from Proposition 2.1 in [4] …”