| Type of Document |
Dissertation |
| Author |
Amaya, Austin J.
|
| Author's Email Address |
amaya@vt.edu |
| URN |
etd-05102012-184739 |
| Title |
Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time Versions |
| Degree |
PhD |
| Department |
Mathematics |
| Advisory Committee |
| Advisor Name |
Title |
| Ball, Joseph A. |
Committee Chair |
| Hagedorn, George A. |
Committee Member |
| Klaus, Martin |
Committee Member |
| Renardy, Michael J. |
Committee Member |
|
| Keywords |
- reproducing kernel Hilbert spaces
- Hardy spaces over left/right half plane
- admissible Sylvester data set
- operator Sylvester equation
- infinite dimensional zero-pole data
- continuous shift semigroups
- Ltwo well-posed linear systems
- continuous-time linear systems
|
| Date of Defense |
2012-04-26 |
| Availability |
unrestricted |
Abstract
Given a full-range simply-invariant shift-invariant subspace M of the vector-valued L2 space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function W so that M may be represented as the image of of the Hardy space H2 on the disc under multiplication by W. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces (M,M×) which together form a direct-sum decomposition of L2. In the case where the pair (M,M×) are finite-dimensional perturbations of the Hardy space H2 and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function W; this realization was parameterized in terms of zero-pole data computed from the pair (M,M×). Later work by Ball-Raney extended this analysis to the case of nonrational functions W 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 (M,M×) of the L2 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.
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Amaya_AJ_D_2012.pdf |
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