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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A problem in convexity leading to the analysis of two functional equations

Author: John V. Ryff
Journal: Trans. Amer. Math. Soc. 305 (1988), 377-396
MSC: Primary 46A55; Secondary 28D05, 39B10, 47B38
MathSciNet review: 920165
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Abstract: Transformation semigroups can often be studied effectively by examining their orbit structure. If the class of transformations has a special quality, such as convexity, it is generally reflected in the orbits. This work is concerned with such a circumstance. The goal is to examine the behavior of transformations on extreme points of orbits through the construction of a class of extreme operators. The construction leads naturally to the study of two functional equations which are analyzed in detail. Information about solutions is obtained through different ${L^2}$-methods depending on whether or not two basic parameters are rational or irrational. In two cases all solutions are classified. In a third an example of a spanning set of solutions is obtained. Techniques of harmonic analysis and ergodic theory are used to study the functional equations.

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Keywords: Functional equations, convexity, extreme points, extreme operators, doubly stochastic, ergodic mappings, measure-preserving transformations
Article copyright: © Copyright 1988 American Mathematical Society