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An Analogue of the Wave Equation and Certain Related Functional Equations

Published online by Cambridge University Press:  20 November 2018

John A. Baker*
Affiliation:
University of Waterloo, Waterloo, Ontario
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Consider the functional equation

(1)

assumed valid for all real x, y and h. Notice that (1) can be written

(2)

a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Aczél, J., Haruki, H., McKiernan, M.A. and Sakovič, G.N., General and regular solutions of functional equations characterizing harmonic polynomials. Aequationes Math. 1 (1968) 3753.Google Scholar
2. Kemperman, J. H. B., A general functional equation. Trans. Am. Math. Soc. 86 (1957) 2856.Google Scholar
3. McKiernan, M.A., Boundedness on a set of positive measure and the mean value property characterizes polynomials on a space Vn. (to appear in Aequationes Mathematicae).Google Scholar
4. Rudin, Walter, Real and Complex Analysis. (McGraw-Hill, New York, 1966).Google Scholar
5. Sakovič, G.N., On d′ Alemberts′ formula for vibrating strings (Russian). Ukrain. Mat. Z. (to appear).Google Scholar
6. Światak, H., On the regularity of the distributional and continuous solutions of the functional equations . Aequationes Math. 1 (1968) 619.Google Scholar
7. Zemanian, A. H., Distribution theory and transform analysis. (McGraw-Hill, New York, 1965).Google Scholar