Controlled weaving frames in Hilbert spaces

Abstract

In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of controlled weaving K-g-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled K-g-woven frames gives ordinary weaving K-frames and vice-versa.

Weaving Continuous Controlled K-g-Gusion Frames in Hilbert Spaces

“…Woven frame is a new notion in frame theory which has been introduced by Bemrose et al [7]. Two frames {f i } i∈I and {g i } i∈I for H are called woven if there exist constants 0 < A ≤ B < +∞ such that for any subset σ ⊂ I the family {f i } i∈σ ∪ {g i } i∈σ c is a frame for H. This frame has been generalized for the discrete as well as the continuous case such as woven fusion frame [17], woven g-frame [24], woven g-fusion frame [25], woven K-g-fusion frame [32], continuous weaving frame [36], continuous weaving fusion frame [33], continuous weaving g-frames [3], weaving continuous K-g-frames [5], controlled weaving frames [29], continuous controlled K-gframes [30] etc. In this paper, woven continuous controlled K-g-fusion frame in Hilbert spaces is presented and some of their properties are going to be established.…”

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].…”

Robustness of controlled fusion frames under erasures of subspaces and the approximation operator

“…Recently, controlled frames have been presented by Balazs and et al in [1] to improve the numerical efficiency of interactive algorithms for inverting the frame operator on Hilbert spaces. After, controlled frames have been studied for another kind of frames in [14,16,17,18]. This manuscript is organized as follows: In Section 2, we study a new identity for the eigenvalues of the controlled frame operator and review the notation of controlled fusion frames with some results about these operators.…”