Redundancy of Fusion Frames in Hilbert Spaces

Abstract

Upon improving and extending the concept of redundancy of frames, we introduce the notion of redundancy of fusion frames, which is concerned with the properties of lower and upper redundancies. These properties are achieved by considering the minimum and maximum values of the redundancy function which is defined from the unit sphere of the Hilbert space into the positive real numbers. In addition, we study the relationship between redundancy of frames (fusion frames) and dual frames (dual fusion frames). Moreover, we indicate some results about excess of fusion frames. We state the relationship between redundancy of local frames and fusion frames in a particular case. Furthermore, some examples are also given.2000 Mathematics Subject Classification. Primary 42C40; Secondary 41A58, 47A58.

Co-compact Gabor Systems on Locally Compact Abelian Groups

“…Assume that {f k } is measurable. It is known that T as in (2.4) defines a bounded linear operator if, and only if, {f k } k∈M is a Bessel family [36]. Hence, the argument in the preceding paragraph shows that {f k } k∈M is a basic frame if, and only if, T as in (2.4) defines a bounded linear operator with im T = K.…”

“…From results in [10,Chapter 3] and [36] we know that this synthesis operator L * ω : L 2 (P ) → ℓ 2 (H ⊥ ) is a well-defined, bounded linear operator if, and only if, the fibers {T g p (ω)} p∈P is a Bessel system. The frame operator L * ω L ω of the family of fibers is called the dual Gramian and is denoted byG ω : ℓ 2 (H ⊥ ) → ℓ 2 (H ⊥ ).…”

Reproducing Pairs of Measurable Functions

“…They have been introduced originally by Ali, Gazeau and one of us [1,2] and also, independently, by Kaiser [13]. Since then, several papers dealt with various aspects of the concept, see for instance [11] or [17]. However, there may occur situations where it is impossible to satisfy both frame bounds.…”

The effect of perturbations of frames and fusion frames on their redundancies

“…In the definition of the redundancy function for fusion frames [22], we consider lower and upper bounds of the normalized version of a fusion frame as lower and upper redundancies.…”