Browsing by Author "Deepshikha"

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    WEAVING FRAMES IN HILBERT SPACES
    (University of Delhi, 2017-10) Deepshikha
    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.