Generalized Translation Operators

Author: McGregor, James Lewin

Year: 1954

Degree: Dissertation (Ph.D.)

Advisor: Bohnenblust, Henri Frederic

Committee Member: Unknown, Unknown

Option: Mathematics; Aeronautics

DOI: 10.7907/799Z-1X12

Abstract

A study is made of generalized translation operators of the Delsarte-Levitan-Povzner type. After reviewing the method of associating such operators with linear second order differential equations, an abstract theory is developed with the aim of constructing an L[subscript 1]-convolution algebra. The chief novelty is a device of comparing one family of translation operators with another "known" family. The Plancherel theorem and Bochner's theorem on positive definite functions are derived by the Krein-Godement method of locally compact group theory. An application to the classical Sturm-Liouville problem is discussed.

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