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

Ryff, John V.

doi:10.1090/S0002-9947-1988-0920165-1

A problem in convexity leading to the analysis of two functional equations
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by John V. Ryff PDF

Trans. Amer. Math. Soc. 305 (1988), 377-396 Request permission

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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