Theses and Dissertations from UMD

Permanent URI for this communityhttp://hdl.handle.net/1903/2

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a give thesis/dissertation in DRUM

More information is available at Theses and Dissertations at University of Maryland Libraries.

Browse

Filters

20111

Settings

Author: search.filters.author.Au-Yeung, EnricoSubject: search.filters.subject.Mathematics

Search Results

Now showing 1 - 1 of 1
  • Thumbnail Image
    Item
    Balayage of Fourier Transforms and the Theory of Frames
    (2011) Au-Yeung, Enrico; Benedetto, John; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Every separable Hilbert space has an orthogonal basis. This allows every element in the Hilbert space to be expressed as an infinite linear combination of the basis elements. The structure of a basis can be too rigid in some situations. Frames gives us greater flexibility than bases. A frame in Hilbert space is a spanning set with the reconstruction property. A frame must satisfy both an upper frame bound and a lower frame bound. The requirement of an upper bound is rather modest. Most of the mathematical difficulty lies in showing the lower bound exists. We examine the theory of Beurling on Balayage of Fourier transforms and the role of spectral synthesis in this theory. Beurling showed that if the condition of Balayage holds, then the lower frame bound for a Fourier frame exists under suitable hypothesis. We extend this theory to obtain lower bound inequalities for other types of frames. We prove that lower bounds exist for generalized Fourier frames and two types of semi-discrete Gabor frames.