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Scales Of Function And Matrix Spaces

Dahma, Alfred (2009) Scales Of Function And Matrix Spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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 (&Omega;,&Sigma;, μ) is a finite, non-atomic measure space, μ a positive measure, then we denote by M(μ) the space of equivalence classes of &Sigma;-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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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairLennard, Christopherlennard@pitt.eduLENNARD
Committee MemberTurett,
Committee MemberBeatrous, Frankbeatrous@pitt.eduBEATROUS
Committee MemberDiestel,
Committee MemberDowling,
Committee MemberPan, Yibiaoyibiao@pitt.eduYIBIAO
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:, etd-08102009-122512
Date Deposited: 10 Nov 2011 19:58
Last Modified: 15 Nov 2016 13:48


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