UMD Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/3

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 given thesis/dissertation in DRUM.

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2007 - 200912010 - 20111

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    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.
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    Wiener's Generalized Harmonic Analysis and Waveform Design
    (2007-02-21) Datta, Somantika; Benedetto, John J; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Bounded codes or waveforms are constructed whose autocorrelation is the inverse Fourier transform of certain positive functions. For the positive function F=1 the corresponding unimodular waveform of infinite length, whose autocorrelation is the inverse Fourier transform of F, is constructed using real Hadamard matrices. This waveform has a autocorrelation function that vanishes everywhere on the integers except at zero where it is one. In this case error estimates have been calculated which suggest that for a pre-assigned error the number (finite) of terms from this infinite sequence that are needed so that the autocorrelation at some non-zero k is within this given error range is `almost' independent of k. In addition, such unimodular codes (both real and complex) whose autocorrelation is the inverse Fourier transform of F=1 has also been constructed by extending Wiener's work on Generalized Harmonic Analysis (GHA) and a certain class of exponential functions. The analogue in higher dimensions is also presented. Further, for any given positive and even function f defined on the integers that is convex and decreasing to zero on the positive integers, waveforms have been constructed whose autocorrelation is f. The waveforms constructed are real and bounded with a bound that depends on the value of f at zero.