Some new inequalities for continuous fusion frames and fusion pairs.

Wei Zhang,Yun-Zhang Li

doi:10.1186/s40064-016-3284-0

Abstract

This paper addresses continuous fusion frames and fusion pairs which are extensions of discrete fusion frames and continuous frames. The study of equalities and inequalities for various frames has seen great achievements. In this paper, using operator methods we establish some new inequalities for continuous fusion frames and fusion pairs. Our results extend and improve ones obtained by Balan, Casazza and Găvruţa.

Highlights

The notion of frame in a general Hilbert space was first introduced by Duffin and Schaeffer in 1952 to study nonharmonic Fourier series (Duffin and Schaeffer 1952)
The study of frame theory has seen great achievements, and frames are widely used in signal processing, quantum measurements, image processing, coding and communication and some other fields (Balan et al 2007; Bownik et al 2015; Casazza 2000; Christensen 2003; Leng and Han 2013; Li and Sun 2008; Li and Zhu 2012; Li et al 2015; Rahimi et al 2006; Strohmer and Heath 2003)
As Casazza, Kutyniok and Li pointed out in Casazza et al (2008), in applications, one is often overwhelmed by a deluge of data assigned to one single frame system, which becomes too large to be handled numerically

Summary

Background

The notion of frame in a general Hilbert space was first introduced by Duffin and Schaeffer in 1952 to study nonharmonic Fourier series (Duffin and Schaeffer 1952). In Faroughi and Ahmadi (2008), the authors defined the continuous fusion frame operator SF : H → H as follows: SF (h) = v2(x)πF(x)(h)dμ(x), ∀h ∈ H. Proposition 4 Let (F , v) be a continuous Bessel fusion mapping on H with bound B, SF1, SF2 are bounded linear operators, and (SF1 )∗(h) = axv2(x)πF(x)(h)dμ(x), ∀h ∈ H. Theorem 5 Let (F , v) and (G, v) be continuous Bessel fusion mappings on H, F and G be a fusion pair, and {ax : x ∈ X} ∈ l∞(X).

Conclusions

Methods