New aspects of weaving K-frames: the excess and duality

Abstract

Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor networks. Likewise weaving K -frames have been proved to be useful during signal reconstructions from the range of a bounded linear operator K. In this paper, we study the notion of weaving and its connection to the duality of K-frames and construct several pairs of woven K-frames. Also, we find a unique biorthogonal sequence for every K-Riesz basis and obtain a $K^*$-frame which is woven by its canonical dual. Moreover, we describe the excess for K-frames and prove that any two woven K-frames in a separable Hilbert space have the same excess. Finally, we introduce the necessary and sufficient condition under which a K-frame and its image under an invertible operator have the same excess.