Dahma, Alfred
(2009)
Scales Of Function And Matrix Spaces.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
The following work is divided into three chapters. In the first chapter, we extend the classical definition of Lebesgue function spaces to include values of p < 0. If (Ω,Σ, μ) is a finite, non-atomic measure space, μ a positive measure, then we denote by M(μ) the space of equivalence classes of Σ-measurable functions. For all p > 0, L−p(µ) is the set M(μ) together with a complete, translation invariant metric, d−p, defined using the decreasing rearrangement of functions ƒ ∈ M(μ). Defined as such, we can extend the inclusion Lq (μ) ⊂ Lp (μ) to all real numbers p and q, with p < q. Furthermore, L−p(μ) can be equipped with an F-norm defined by ||f|| = d−p(f, 0). The second chapter deals with the theory of Hilbert frames. We prove several inequalities relating the Schatten norm of the frame operator, S, to the p-norms of the frame elements, ƒj. This is done first in finite dimensional Hilbert spaces, then extended to infinite dimensions using a truncated frame operator for finite subsets of the frame. In the final section of this chapter, we construct a frame for which the averaged 1-norm of the associated Gram matrix exhibits an optimal growth rate. In the paper Generalized Roundness and Negative Type, Lennard, Tonge, and Westonshow that the geometric notion of generalized roundness in a metric space is equivalent tothat of negative type. Using this equivalent characterization, along with classical embedding theorems, the authors prove that for p > 2, L p fails to have generalized roundness q for any q > 0. It is noted, as a consequence, that the Schatten class C p, for p > 2, has maximal generalized roundness 0. In the third chapter, we prove that this result remains true for p in the interval (0, 2).
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
30 September 2009 |
| Date Type: |
Completion |
| Defense Date: |
30 July 2009 |
| Approval Date: |
30 September 2009 |
| Submission Date: |
10 August 2009 |
| Access Restriction: |
5 year -- Restrict access to University of Pittsburgh for a period of 5 years. |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
generalized roundness; frames; schatten class |
| Other ID: |
http://etd.library.pitt.edu/ETD/available/etd-08102009-122512/, etd-08102009-122512 |
| Date Deposited: |
10 Nov 2011 19:58 |
| Last Modified: |
15 Nov 2016 13:48 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/9040 |
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