Duals and multipliers of K -fusion frames
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Duals and multipliers of K -fusion frames Mitra Shamsabadi1 · Ali Akbar Arefijamaal1
· Ghadir Sadeghi1,2
Received: 17 October 2019 / Revised: 18 March 2020 / Accepted: 21 June 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we introduce the concept of K -fusion frames and propose the duality for such frames. The relation between the local frames of K -fusion frames with their duals is studied. The elements from the range of a bounded linear operator K can be reconstructed by K -frames. Also, we establish K -fusion frame multipliers and investigate reconstruction of the range of the operator K by them. Keywords Fusion frame · K -fusion frame · K -dual · Multiplier Mathematics Subject Classification Primary 41A58; Secondary 42C15
1 Introduction, notation and motivation The theory of frames plays an important role in wavelet theory [17] as well as (timefrequency) analysis for functions in L 2 (Rd ) [10]. The traditional applications of frames are signal processing [7,17], image processing [8], sampling theory [14] and communication [17], moreover, recently the use of frames also in numerical analysis for the solution of operator equation by adaptive schemes is investigated [11]. Frame multipliers have many interesting features in psychoacoustical modeling and denoising [5,19]. Also, several extensions of multipliers are proposed in [4,21,24]. It is important to represent the inverse of a multiplier if it exists [6,23]. From a theoretical point of view, every multiplier is a natural generalization of the frame operator. Hence, it
B
Ali Akbar Arefijamaal [email protected]; [email protected] Mitra Shamsabadi [email protected] Ghadir Sadeghi [email protected]
1
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran
2
Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran
M. Shamsabadi et al.
is worthwhile to discuss its invertibility. Multipliers have application as time-variant filters [5,12,20] in acoustical signal processing. Recently, K - fusion frames are introduced [1,17,18]. Our purpose is to introduce K - fusion frame multipliers and apply them to reconstruct elements from the range of a bounded operator K . We also discuss the invertibility of K-fusion frame multipliers. For two sequences := {φi }i∈I and := {ψi }i∈I in a Hilbert space H and a sequence m = {m i }i∈I of complex scalars, the operator Mm,, : H → H given by Mm,, f =
m i f , ψi ϕi , ( f ∈ H),
(1.1)
i∈I
is called a multiplier. The sequence m is called symbol. If and are Bessel sequences for H and m ∈ ∞ , then Mm,, is a well-defined bounded operator and Mm,, ≤ √ B B m∞ , where B and B are Bessel bounds of and , respectively [3]. The invertibility of multipliers, which plays a key role in the topic, is discussed in [3,6,23]. K -frames which recently introduced by Gˇavru¸ta are generalization of frames, in the meaning that the lower frame bound only holds for that admits to reco
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