Sums of Hilbert space frames

Abstract

We give simple necessary and sufficient conditions on Bessel sequences { f i } and {g i } and operators L 1 , L 2 on a Hilbert space H so that {L 1 f i + L 2 g i } is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames.

Applications of Parameterizations of Oblique Duals of a Frame Sequence

“…As can be seen in Definition 2.4, an oblique dual of a frame sequence is a natural symmetric dual of the frame sequence. Hence, this result on frame sequences contrasts with the result on the sum of a frame and one of its dual frame by Obeidat et al [19] mentioned previously.…”

“…Possibly inspired by this result, Obeidat, Samarah, Casazza, and Tremain showed that, among other things, if F is a frame for and G is any one of its (alternate or generalized) dual frames (for ), then F + G is also a frame for [19]. This implies that F + G is a frame sequence if F is a frame sequence and G is a type I dual of F .…”

“…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]. It was proved by Obeidat, Samarah, Casazza, and Tremain that the sum of a frame sequence and one of its type I duals is also a frame sequence [19]. We first show that the sum of a frame sequence and one of its type Parameterizations of Oblique Duals 897 II duals is a frame sequence and give necessary conditions for the range space of the sum, and then we apply the parameterization by Christensen and Eldar to show that the sum of a frame sequence and one of its oblique duals may not be a frame sequence.…”

More on Sums of Hilbert Space Frames

“…Therefore L is invertible and so {f i } i∈I is a frame for H with the frame operator L −1 S L (L * ) −1 . Here, we also show that the equivalence of part (1) and (2) in Proposition 3.1 of [3], is not true in general. Indeed, if T 1 L * 1 + T 2 L * 2 is an invertible operator, then {L 1 f i + L 2 g i } i∈I is a frame for H but the inverse is not true.…”

Constructions of Frames by Disjoint Frames

“…There are many reference articles on such problems such as [11][12][13][14][15]. In [16], the authors studied the sums of Hilbert space frames, which is a simple way to construct a large number of new frames from existing frames, where just sums of two frames are considered. In this article, we will apply the techniques established in [4] to discuss the constructing problem.…”