More on Inequalities for Weaving Frames in Hilbert Spaces

Zhong-Qi Xiang

doi:10.3390/math7020141

Abstract

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

Highlights

Throughout this paper, H is a separable Hilbert space, and IdH is the identity operator on H.The notations J, R, and B(H) denote, respectively, an index set which is finite or countable, the real number set, and the family of all linear bounded operators on H.A sequence F = { f j } j∈J of vectors in H is a frame if there are constants A, B > 0 such thatAk x k2 ≤ ∑ |h x, f j i|2 ≤ Bk x k2, ∀ x ∈ H. (1) j ∈JThe frame F = { f j } j∈J is said to be Parseval if A = B = 1
Motivated by a problem deriving from distributed signal processing, Bemrose et al [23] put forward the notion of weaving frames for Hilbert spaces
Parseval frames identity arising in their work on effective algorithms for computing the reconstructions of signals, which was extended to general frames and alternate dual frames [32], and based on the work in [31,32], some inequalities for generalized frames associated with a scalar are established

Summary

Introduction

Throughout this paper, H is a separable Hilbert space, and IdH is the identity operator on H. Motivated by a problem deriving from distributed signal processing, Bemrose et al [23] put forward the notion of (discrete) weaving frames for Hilbert spaces. Parseval frames identity arising in their work on effective algorithms for computing the reconstructions of signals, which was extended to general frames and alternate dual frames [32], and based on the work in [31,32], some inequalities for generalized frames associated with a scalar are established (see [33,34,35]). We recall that a frame H = {h j } j∈J is said to be an alternate dual frame of { f j } j∈σ ∪ { g j } j∈σc if x=. ∑ hx, hjifj + ∑c hx, h j i gj j∈σ j∈σ are well-defined and, further, SF GH , SHF G ∈ B(H)

Main Results and Their Proofs

We also get

On the other hand we get σc c