SOME RESULTS OF K-FRAMES AND THEIR MULTIPLIERS

2020 Turkish Journal of Mathematics  
K -frames are strong tools for the reconstruction of elements from range of a bounded linear operator K on a 4 separable Hilbert space H . In this paper, we study some properties of K -frames and introduce the K -frame multipliers. 5 We also focus on representing elements from the range of K by K -frame multipliers. 6 Key words: K -frame, K -dual, K -left inverse, K -right inverse, multiplier 7 1. Introduction, notation and motivation 8 For the first time, frames in Hilbert space were offered
more » ... pace were offered by Duffin and Schaeffer in 1952 and were brought to life 9 by Daubechies, Grossman and Meyer [18] . A frame allows each element in the underlying space to be written as 10 a linear combination of the frame elements, but linear independence between the frame elements is not required. 11 This fact has key role in applications as signal processing, image processing, coding theory and more. For more 12 details and applications of ordinary frames see [2, 3,[7][8][9][10][11][12][13][14][15][16] 20]. K -frames which recently introduced by Gǎvruţa 13 are a generalization of frames, in the meaning that the lower frame bound only holds for range of a linear and 14 bounded operator K in a Hilbert space [19]. 15 A sequence m := {m i } i∈I of complex scalars is called semi-normalized if there exist constants a and b 16 such that 0 < a ≤ |m i | ≤ b < ∞ , for all i ∈ I . For two sequences Φ := {ϕ i } i∈I and Ψ := {ψ i } i∈I in a Hilbert 17 space H and a sequence m of complex scalars, the operator M m,Φ,Ψ : H → H given by 18 M m,Φ,Ψ is called a multiplier. The sequence m is called the symbol. If Φ and Ψ are Bessel sequences for H and B Ψ 20 are Bessel bounds of Φ and Ψ , respectively [4]. Frame multipliers have many applications in psychoacoustical 21 modeling and denoising [6, 24]. Also, several generalizations of multipliers are proposed [5, 21, 22]. 22 It is important to detect the inverse of a multiplier if it exists [8, 23]. Our aim is to introduce K -frame 23 multipliers and apply them to reconstruct elements from the range of K . 24 Throughout this paper, we suppose that H is a separable Hilbert space, I a countable index set and 25 I H the identity operator on H . For two Hilbert spaces H 1 and H 2 we denote by B(H 1 , H 2 ) the collection of 26
doi:10.3906/mat-1912-95 fatcat:bthzvqg3xjda5fiq633zgf7i6u