Manufacturing Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces
- PDF / 334,471 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 69 Downloads / 123 Views
Manufacturing Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces Fahimeh Arabyani-Neyshaburi1 · Ali Akbar Arefijamaal1 Received: 27 October 2019 / Revised: 11 April 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract Weaving Hilbert space frames have been introduced recently by Bemrose et al. to deal with some problems in distributed signal processing. In this paper, we survey this topic from the viewpoint of the duality principle. In this regard, not only we obtain new properties in weaving frame theory related to dual frames but also we bring up new approaches for manufacturing pairs of woven frames. Specifically, we give some sufficient conditions under which a frame with its canonical dual, alternate duals or approximate duals constitute some concrete pairs of woven frames. Moreover, we provide some approaches for constructing weaving frames by using small perturbations and present a condition where different operators preserve the weaving property. As a consequence, the canonical duals of two woven frames are woven. Keywords Riesz bases · Weaving Hilbert space frames · Duality principle Mathematics Subject Classification 42C15
1 Introduction Today, frame theory has been developed in various areas of pure and applied science and engineering [3,7,9,10,12]. Due to redundance requirement in applications and theoretical goals, in over the past two decades, several generalizations of classical frames have been presented [1,2,4,13,17–20,22,23]. Recently, Bemrose et al. [6] have introduced a new concept in frame theory due to some problems in distributed sig-
Communicated by Sorina Barza.
B
Ali Akbar Arefijamaal [email protected] Fahimeh Arabyani-Neyshaburi [email protected]
1
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran
123
F. Arabyani-Neyshaburi, A. A. Arefijamaal
nal processing, called weaving Hilbert space frames. This notion also has potential applications in wireless sensor networks which require distributed processing using different frames. Two given frames {ϕi }i∈I and {ψi }i∈I for a separable Hilbert space H are woven if there exist constants 0 < A ≤ B < ∞ so that for every subset J ⊂ I , the family {ϕi }i∈J ∪ {ψi }i∈J c is a frame for H with frame bounds A and B. In [6,14], the authors presented some remarkable properties of weaving frames and weaving Riesz bases. In this paper, we give new basic properties of weaving frames related to dual frames to survey under which conditions a frame with its dual constitute woven frames. Moreover, we present some approaches for constructing concrete pairs of woven frames. The paper is organized as follows. In Sect. 2, we will review some basic definitions and results related to frames and weaving frames in Hilbert spaces. Section 3 is devoted to presenting several methods of constructing concrete pairs of woven frames. We also obtain some conditions to prove that a frame with its duals or approximate duals constitutes some pairs of wove
Data Loading...
