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Projective actions, invariant sigma-curves and quadratic functional equations

Published online by Cambridge University Press:  24 October 2008

Patrick J. McCarthy
School of Mathematical Sciences, Queen Mary College, London E1 4NS


The quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.

Research Article
Copyright © Cambridge Philosophical Society 1985

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