P-woven dual frames
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P-woven dual frames Akram Bibak Hafshejani1 · Mohammad Ali Dehghan1 Received: 31 August 2019 / Revised: 31 August 2019 / Accepted: 17 April 2020 © Springer Nature Switzerland AG 2020
Abstract The present study deals with the recently introduced concept of P-woven frames. Two frames {ϕi }i∈I and {ψi }i∈I for a Hilbert space H are called P-woven, if there exists a nontrivial subset σ of I such that the family {gi }i∈I represented as the triple ({ϕi } , {ψi } , σ ) is a frame defined by gi = ϕi if i ∈ σ and gi = ψi , otherwise. The duals of frames have an essential role in the reconstruction of vectors (or signals) in terms of the frame elements. In this paper we will prove some results in regards to P-woven property of the duals of P-woven frames. We then introduce the concept of P-woven duals of a collection of frames and obtain some necessary and sufficient conditions in which a collection of frames have P-woven duals. Keywords Dual frames · P-woven frames · P-woven dual frames Mathematics Subject Classification Primary 42C15; Secondary 47B65 · 15A03
1 Introduction and preliminaries In this section we give the necessary background for the paper. The reader is referred to [5,7] for more details and the main concepts of frame theory. Throughout this paper, H is a separable Hilbert space and I is a subset of natural numbers N, and “partition of a set” is one having no empty part. A collection F = { f i }i∈I of vectors in H is called a frame for H if there exist constants 0 < A ≤ B < ∞ such that for all x ∈ H , A x2 ≤
|x, f i |2 ≤ B x2 ,
i∈I
B
Akram Bibak Hafshejani [email protected] Mohammad Ali Dehghan [email protected]
1
Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 518, Rafsanjan, Iran
A. Bibak Hafshejani, M. A. Dehghan
the constants A and B are called lower and upper frame bounds, respectively. If only the second part of the above inequality holds, then it is called a Bessel sequence with Bessel bound B. If A = B, it is called a tight frame. The set of all elements of a frame is called range of the frame. If a frame ceases to be a frame when an arbitrary element is removed, it is called an exact frame. The pre-frame operator or the synthesis operator T for a Bessel sequence F = { f i }i∈I in H is a bounded linear operator from 2 (I ) to H defined by T {ci }i∈I = i∈I ci f i . The adjoint operator T ∗ of T defined by T ∗ x = {x, f i }i∈I is called the analysis operator. When F is a frame for H , the operator S = T T ∗ is the frame operator of F, and we have Sx =
x, f i f i ,
i∈I
for every x ∈ H . The frame operator S is bounded, positive, self-adjoint and invertible. A Riesz basis for H is a sequence of the form {U ei }i∈I , where {ei }i∈I is an orthonormal basis for H and U : H → H is a bounded bijective operator. A frame which is a Schauder basis is called a Riesz basis. A frame which is not a Riesz basis is said to be overcomplete. The concept of weaving frames was introduced by Bemrose, Casazza, Gröchenig, Lammers and Lynch [1] which are powerful to
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