WEAVING FRAMES IN HILBERT SPACES
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Date
2017-10
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University of Delhi
Abstract
The work in the thesis entitled “Weaving Frames in Hilbert Spaces” deals with
the study of weaving properties of different types of frames in separable Hilbert spaces.
This is motivated by a new concept of “weaving frames” in separable Hilbert spaces;
introduced by Bemrose, Casazza, Gr¨ochenig, Lammers and Lynch [13] with an eye on
various applications of frames in distributed signal processing. Let I be a countable
set. Two discrete frames {fi}i∈I and {gi}i∈I
for a separable Hilbert space H are said
to be woven if there are universal positive constants A and B such that for every
subset σ ⊂ I, the family {fi}i∈σ ∪ {gi}i∈σc is a frame for H with lower and upper
frame bounds A and B, respectively. We say that {fi}i∈I and {gi}i∈I are weakly
woven if for every subset σ ⊂ I, the family {fi}i∈σ ∪ {gi}i∈σc is a frame for H.
Weaving frames has potential applications in wireless sensor networks that require
distributed processing under different frames, as well as pre-processing of signals
using Gabor frames. Casazza and Lynch in [30] reviewed fundamental properties of
weaving frames. They also discussed weaving equivalent of an unconditional basis.
Fundamental properties of weaving fusion frames can be found in [56]. Casazza,
Freeman and Lynch [31] extended the concept of weaving Hilbert space frames to the
Banach space setting. We study weaving properties of ordinary frames, generalized
frames and continuous frames in separable Hilbert spaces. The contents of the thesis
have been published/accepted/communicated in [55, 57, 58, 101, 102]. The thesis is
apportioned into five chapters. Chapter 1 provides a brief review of frames and Riesz
bases in separable Hilbert spaces. Chapters 2 to 5 are the embodiment of our study
in the thesis.