A problem in convexity leading to the analysis of two functional equations
HTML articles powered by AMS MathViewer
- 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.References
- Jean Dhombres, Some aspects of functional equations, Lecture Notes, Chulalongkorn University, Department of Mathematics, Bangkok, 1979. With a Thai preface by Wirun Bu nsambatΓ. MR 550292
- Joram Lindenstrauss, A remark on extreme doubly stochastic measures, Amer. Math. Monthly 72 (1965), 379β382. MR 181728, DOI 10.2307/2313497
- A. RΓ©nyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477β493. MR 97374, DOI 10.1007/BF02020331
- John V. Ryff, On the representation of doubly stochastic operators, Pacific J. Math. 13 (1963), 1379β1386. MR 163171
- John V. Ryff, Orbits of $L^{1}$-functions under doubly stochastic transformations, Trans. Amer. Math. Soc. 117 (1965), 92β100. MR 209866, DOI 10.1090/S0002-9947-1965-0209866-5
- John V. Ryff, The functional equation $af(ax)+bf(bx+a)=bf(bx)+af(ax+b):$ extensions and almost periodic solutions, Bull. Amer. Math. Soc. 82 (1976), no.Β 2, 325β327. MR 404903, DOI 10.1090/S0002-9904-1976-14043-8
- John V. Ryff, The functional equation $F(ax)+F(bx+a)=F(bx)+F(ax+b)$, entire and almost periodic solutions, J. Reine Angew. Math. 302 (1978), 116β136. MR 511697, DOI 10.1515/crll.1978.302.116
- YΕ«ji Sakai and Tetsuya Shimogaki, Equimeasurability of functions and doubly stochastic operators, K\B{o}dai Math. Sem. Rep. 24 (1972), 203β211. MR 308831
- Ray C. Shiflett, Extreme Markov operators and the orbits of Ryff, Pacific J. Math. 40 (1972), 201β206. MR 302865
- Peter Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121β153. MR 466493, DOI 10.1090/S0002-9947-1978-0466493-1
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 305 (1988), 377-396
- MSC: Primary 46A55; Secondary 28D05, 39B10, 47B38
- DOI: https://doi.org/10.1090/S0002-9947-1988-0920165-1
- MathSciNet review: 920165