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(Unpublished)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, nonatomic measure space, μ a positive measure, then we denote by M(μ) the space of equivalence classes of Σmeasurable 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 Fnorm 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 pnorms 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 1norm 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Unpublished 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: 

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/etd08102009122512/, etd08102009122512 
Date Deposited: 
10 Nov 2011 19:58 
Last Modified: 
15 Nov 2016 13:48 
URI: 
http://dscholarship.pitt.edu/id/eprint/9040 
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