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Solution of the ‘Cube’ Functional Equation in Terms of ‘Trilinear Coefficients’

Published online by Cambridge University Press:  20 November 2018

H. T. Hemdan
H. T. Hemdan*
Affiliation:
Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada
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We consider the following three functional equations

1

2

3

where f:R3→R.

Considering their geometric meaning, equations (1) and (2) are known as ‘Cube’ and ‘Octahedron’ functional equations, respectively. Under the assumption of continuity, Haruki [2] has proved that (1) and (2) are equivalent. Etigson [3], has proved the equivalence of (1) and (2) under no regularity assumption. We will give here another proof. Also, under the assumption of continuity, Haruki has solved the ‘Cube’ functional equation. He gave the solution as a certain polynomial of fifth degree in x, y, z individually whose terms are the partial derivatives of a given polynomial.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 2 , 01 June 1976 , pp. 181 - 191
Copyright
Copyright © Canadian Mathematical Society 1976

Footnotes

(1)

Presented at the 34th Ontario Mathematics Meeting, University of Guelph, February 1975.

References

1. Aczél, J., Haruki, H., McKiernan, M. A., Sakovic, G. N., General and Regular Solutions of Functional Equations Characterizing Harmonic Polynomials, Aequationes Mathematicae 1 (1968), 37–53.CrossRefGoogle Scholar
2. Haruki, H., On a ‘Cube’ Functional Equation, Aequationes Mathematicae 3 (1969), 156–159.CrossRefGoogle Scholar
3. Etigson, L., Equivalence of ‘Cube’ and ‘Octahedron’ Functional Equations, Aequationes Mathematicae 10 (1974), 50–65.CrossRefGoogle Scholar