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Graphs in the plane invariant under an area preserving linear map and general continuous solutions of certain quadratic functional equations

Published online by Cambridge University Press:  24 October 2008

P. J. McCarthy
Affiliation:
School of Mathematical Sciences, Queen Mary College, London El 4NS
M. Crampin
Affiliation:
Faculty of Mathematics, The Open University, Milton Keynes MK7 6AA
W. Stephenson
Affiliation:
Department of Mathematics, University College, London WC1E 6BT

Abstract

The requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the form

for all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

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