Abstract

Hedenmalm, Lindqvist, and Seip in 1997 [11] revitalized the modern study of Dirichletseries by defining the space H 2 and considering it as isometrically isomorphic to the Hardyspace of the infinite polytorus H 2 (T ∞ ). This allowed a new viewpoint to be applied toclassical theorems, including Carlson’s theorem about the integral in the mean of a Dirichletseries. Carlson’s theorem holds only for vertical lines in the right half plane, and cannot beextended to the boundary in full generality (as shown by Saksman and Seip in [14]). However,Carlson’s theorem can be shown to hold on the imaginary axis for a more restrictive class ofDirichlet series, and we shall do so.The main result contained in this dissertation is a generalized version of Carlson’s the-orem: given a Borel probability measure on the polytorus, a measure is constructed on theimaginary axis so that the integral in the mean is equal to the integral on the polytorus.Chapter 1 contains background material on Dirichlet series, including questions of con-vergence, the Bohr lift, and spaces of Dirichlet series. Chapter 2 is the statement and proofof the main result generalizing Carlson’s theorem.Finally, Chapter 3 is a small result about weighted spaces of Dirichlet series from workdone jointly with Houry Melkonian.

Committee Chair

John E. McCarthy

Committee Members

Gregory Knese, Constanze Liaw, Xiang Tang, Brett Wick,

Comments

Permanent URL: https://doi.org/10.7936/K7BK1BS1

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Spring 5-15-2018

Language

English (en)

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Mathematics Commons

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