Frame-related operators for woven frames

Abstract

A new notion in frame theory has been introduced recently under the name woven-weaving frames by Bemrose et. al. In the studying of frames, some operators like analysis, synthesis, Gram and frame operator play the central role. In this paper, for the first time, we introduce and define these operators for woven-weaving frames and review some properties of them. In continuation, we investigate the effect of different types of operators on the woven frames and their bounds. Also, we provide some conditions that shows sum of woven frames are also woven frames. Finally, we apply these properties with an example. IntroductionThe theory of frames plays an important role in signal processing because of their resilience to quantization [15], resilience to additive noise, as well as their numerical stability of reconstruction and greater freedom to capture signal characteristics. Also frames have been used in sampling theory to oversampled perfect reconstruction filter banks, system modeling, neural networks and quantum measurements [12]. New applications in image processing, robust transmission over the internet and wireless [16], coding and communication [23] were given.Discrete frames in Hilbert spaces has been introduced by Duffin and Schaeffer [11] and popularized by Daubechies, Grossmann and Meyer [10]. A discrete frame is a countable family of elements in a separable Hilbert space which allows stable and not necessarily unique decompositions of arbitrary elements in an expansion of frame elements.The last two decades have seen tremendous activity in the development of frame theory and many generalizations of frames have come into existence which include bounded quasi-projectors [13], fusion frames [5], pseudoframes [19], oblique frames [8], g-frames [24], continuous frames [21], Kframes [14], fractional calculus [17, 18], Hilbert-Schmidt frames [22] and etc.In one of the direction of applications of frames in signal processing, a new concept of woven-weaving frames in a separable Hilbert space introduced by

Woven Frames in Quaternionic Hilbert Spaces

“…Furthermore, woven frames have inherent applications in wireless sensor networks which are based upon distributed processing under family of frames. For more literature on woven frame, one may refer to [19][20][21].…”

Woven (Weaving) Frames in Banach Spaces

“…Inspired by a problem raised in distributed signal processing, Bemrose et al [1] introduced the concept of weaving frames in separable Hilbert spaces and observed that the weaving frames may be applied in sensor networks which requires distributed processing under different frames. In recent years, a considerable amount of research has been conducted to extend the notion of weaving frames to different settings which include weaving frames in Banach spaces, continuous weaving frames, generalized weaving frames, weaving Riesz bases, weaving fusion frames, weaving controlled frames and weaving vector-valued frames [5,6,20,22,[24][25][26][31][32][33][34].…”

More on Inequalities for Weaving Frames in Hilbert Spaces

“…The theory may be applied to deal with wireless sensor networks that require distributed processing under different frames, which could also be used in the pre-processing of signals by means of Gabor frames. Recently, weaving frames have attracted many scholars' attention, please refer to [24][25][26][27][28][29][30] for more information.…”