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Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time Versions

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dc.contributor.authorAmaya, Austin J.en
dc.contributor.committeechairBall, Joseph A.en
dc.contributor.committeememberHagedorn, George A.en
dc.contributor.committeememberKlaus, Martinen
dc.contributor.committeememberRenardy, Michael J.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T20:11:50Zen
dc.date.adate2012-05-30en
dc.date.available2014-03-14T20:11:50Zen
dc.date.issued2012-04-26en
dc.date.rdate2012-05-30en
dc.date.sdate2012-05-10en
dc.description.abstractGiven a full-range simply-invariant shift-invariant subspace <i>M</i> of the vector-valued <i>L<sup>2</sup></i> space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function <i>W</i> so that <i>M</i> may be represented as the image of of the Hardy space <i>H<sup>2</sup></i> on the disc under multiplication by <i>W</i>. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces <i>(M,M<sup>Ã </sup>)</i> which together form a direct-sum decomposition of <i>L<sup>2</sup></i>. In the case where the pair <i>(M,M<sup>Ã </sup>)</i> are finite-dimensional perturbations of the Hardy space <i>H<sup>2</sup></i> and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function <i>W</i>; this realization was parameterized in terms of zero-pole data computed from the pair <i>(M,M<sup>Ã </sup>)</i>. Later work by Ball-Raney extended this analysis to the case of nonrational functions <i>W</i> where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs <i>(M,M<sup>Ã </sup>)</i> of the <i>L<sup>2</sup></i> spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans.en
dc.description.degreePh. D.en
dc.identifier.otheretd-05102012-184739en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-05102012-184739/en
dc.identifier.urihttp://hdl.handle.net/10919/27636en
dc.publisherVirginia Techen
dc.relation.haspartAmaya_AJ_D_2012.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectreproducing kernel Hilbert spacesen
dc.subjectHardy spaces over left/right half planeen
dc.subjectadmissible Sylvester data seten
dc.subjectoperator Sylvester equationen
dc.subjectinfinite dimensional zero-pole dataen
dc.subjectcontinuous shift semigroupsen
dc.subjectLtwo well-posed linear systemsen
dc.subjectcontinuous-time linear systemsen
dc.titleBeurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time Versionsen
dc.typeDissertationen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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