Weaving information packets

Abstract

The concept of weaving of frames for Hilbert spaces was introduced by Bemrose et al. in 2016. Two frames ${\left\{f_k\right\}}_{k\in I},{\left\{g_k\right\}}_{k\in I}$ are woven if the “mixed system” ${\left\{f_k\right\}}_{k\in \sigma }\cup {\left\{g_k\right\}}_{k\in I\setminus \sigma }$ is a frame for each index set $\sigma \subset I;$ that is, processing a signal using two woven frames yields a certain stability against loss of information. The concept easily extends to $N$ frames, for any integer $N>2.$ Unfortunately it is nontrivial to construct useful woven frames, and the literature is sparse concerning explicit constructions. In this paper we introduce so-called information packets, which contain as well frames as fusion frames as special case. The concept of woven frames immediately generalizes to information packets, and we demonstrate how to construct practically relevant woven information packets based on particular wavelet systems in $L^2\left(R\right).$ Interestingly, we show that certain wavelet systems can be split into $N$ woven information packets, for any integer $N\ge 2.$ We finally consider corresponding questions for Gabor system in $L^2\left(R\right),$ and prove that for any fixed $N\in N$ we can find a Gabor frame that can be split into $N$ woven information packets; however, in contrast to the wavelet case, the density conditions for Gabor system excludes the possibility of finding a single Gabor frame that works simultaneously for all $N\in N.$