Finite Fourier transform approximation and Riemann sum approximation for functions that decay in time and frequency.
Item
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Title
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Finite Fourier transform approximation and Riemann sum approximation for functions that decay in time and frequency.
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Identifier
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AAI9605622
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identifier
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9605622
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Creator
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Litwin, Jeffrey.
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Contributor
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Adviser: Louis Auslander
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Date
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1995
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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For functions with finite time and frequency energy moments, we find upper bounds for the error of finite Fourier transform approximation to the Fourier transform. The error can be measured as the maximum error over all of the points of the FFT, or using a discrete {dollar}{lcub}\bf L{rcub}\sp2{dollar} distance, or using a continuous {dollar}{lcub}\bf L{rcub}\sp2{dollar} distance.;Using the machinery developed, we also find an upper bound for the error of approximating the {dollar}{lcub}\bf L{rcub}\sp2{dollar} norm of a function by a Riemann sum. From this result, an upper bound is also derived for the error of approximating the {dollar}{lcub}\bf L{rcub}\sp2{dollar} inner product, as well as the error of approximating the integral of an {dollar}{lcub}\bf L{rcub}\sp1{dollar} function, by a Riemann sum.;As an application of the {dollar}{lcub}\bf L{rcub}\sp2{dollar} norm approximation theorem, we prove an analog of the Landau-Pollak-Slepian approximate dimension theorems for a certain set of functions that is approximately time-and-bandlimited for large duration N and bandwidth M. This set can be approximately parameterized with N M parameters, with the error approaching zero as N M approaches infinity.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
