Weaving properties of generalized continuous frames generated by an iterated function system

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

show abstract

“…After that, Vashisht, Deepshikha, Arefijamaal and etc. have done more studies over the past few years [1,9,26,27].…”

Section: 2mentioning

confidence: 99%

Frame-related operators for woven frames

Rahimi

Samadzadeh

Daraby

2019

Int. J. Wavelets Multiresolut Inf. Process.

View full text Add to dashboard Cite

show abstract

“…Various necessary conditions were presented for the weaving of infinitely many frames in Hilbert spaces. The concept of weaving discrete frames was extended to weaving continuous frames in [12,13]. Weaving properties of Θ-g-frames, g-frames, and fusion frames can be found in [14,15,16].…”

Section: Weaving Framesmentioning

confidence: 99%

Weaving K-Frames in Hilbert Spaces

Deepshikha

Vashisht

2018

Results Math

View full text Add to dashboard Cite

Generalized frames (g-frames) are natural generalization of standard frames whereas generalized frames for operators (Θ-g-frames) are g-frames in which the lower frame inequality is controlled by a linear bounded operator Θ on a Hilbert space. In this article, we study weaving properties of g-frames and Θ-g-frames. We present some sufficient conditions for Θ-g-frames to be woven. It is shown that the family of Θ-g-frames obtained by applying bounded operators (whose adjoint operator is surjective) to a family of weaving Θ-g-frames is woven. A class of weaving generalized frames is established by generalized frame operator. A Paley-Weiner type perturbation result for weaving Θ-g-frames is presented. Finally, we show that weaving g-Riesz bases need not behave same as weaving ordinary Riesz bases.

show abstract

“…For application of frames in both pure and applied mathematics, we refer to books of Casazza and Kutyniok [3], Christensen [5], Han [16], Heil [17] and Krivoshein, Protasov and Skopina [19]. Nowadays, the theory of iterated function systems, quantum mechanics and wavelets are emerging in important applications in the frame theory, see [12,22,24]. A very recent work on discrete frames of translates and discrete wavelet frames and their duals in finite dimensional spaces can be found in [10,11].…”

Section: Introductionmentioning

confidence: 99%

Unitary Extension Principle for Nonuniform Wavelet Frames in L2(ℝ)

Malhotra¹,

Vashisht²

2021

Z. mat. fiz. anal. geom.

View full text Add to dashboard Cite

Parseval frames have attracted attention of engineers and physicists due to their potential applications in signal processing. In this paper, we study the construction of nonuniform Parseval wavelet frames for the Lebesgue space L 2 (R), where the related translation set is not necessary a group. The main purpose of this paper is to prove the unitary extension principle (UEP) and the oblique extension principle (OEP) for the construction of multigenerated nonuniform Parseval wavelet frames for L 2 (R). Some examples are also given to illustrate the results.

show abstract

Weaving properties of generalized continuous frames generated by an iterated function system

Cited by 36 publications

References 6 publications

Frame-related operators for woven frames

Frame-related operators for woven frames

Weaving K-Frames in Hilbert Spaces

Unitary Extension Principle for Nonuniform Wavelet Frames in L2(ℝ)

Contact Info

Product

Resources

About