The Weil transform and ambiguity functions.
Item
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Title
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The Weil transform and ambiguity functions.
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Identifier
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AAI9325098
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identifier
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9325098
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Creator
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Geshwind, Frank B.
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Contributor
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Adviser: Louis Auslander
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Date
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1993
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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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Recall that the Weil transform, {dollar}\Theta,{dollar} is an intertwining operator between the {dollar}{lcub}\bf L{rcub}\sp2{dollar} Dirac representation D, of the real, 3-dimensional Heisenberg group, {dollar}{lcub}\cal N{rcub},{dollar} and the regular representation of {dollar}{lcub}\cal N{rcub}{dollar} on {dollar}{lcub}\bf L{rcub}\sp2(\Gamma\\{lcub}\cal N{rcub}),{dollar} where {dollar}\Gamma{dollar} is the integer Heisenberg group. As such, it provides a unitary isomorphism between {dollar}{lcub}\bf L{rcub}\sp2({lcub}\bf R{rcub}){dollar} and {dollar}{lcub}\bf L{rcub}\sp2(\Gamma\\{lcub}\cal N{rcub}).{dollar}.;In this work, the definition of {dollar}\Theta{dollar} is extended so that its domain includes various types of distributions. Results in the literature relating smoothness and moments of functions on R to smoothness and moments of the corresponding functions on {dollar}\Gamma\\{lcub}\cal N{rcub},{dollar} are extended to distributions.;A study of the local structure of smooth functions in {dollar}\Theta({lcub}\bf L{rcub}\sp2({lcub}\bf R{rcub})){dollar} is carried out. A product theorem is given for certain functions which we describe as having smooth growth. Theorems are given about the local homology of these functions. These results are applied to the problem of dividing smooth functions in {dollar}\Theta({lcub}\bf L{rcub}\sp2({lcub}\bf R{rcub})){dollar} by each other, which is related to questions of stable Weyl-Heisenberg wavelet expansions for such functions.;The Weil transform and the above insights into smooth functions in {dollar}\Theta({lcub}\bf L{rcub}\sp2({lcub}\bf R{rcub})){dollar} are applied to the thumbtack synthesis problem for radar ambiguity functions. We study known constructions for this problem, in the present context, and then provide new constructions using the insights gained.
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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.
