An Analogue of the Wave Equation and Certain Related Functional Equations

John A. Baker

doi:10.4153/CMB-1969-109-6

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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].

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) 37–53.Google Scholar

2. Kemperman, J. H. B., A general functional equation. Trans. Am. Math. Soc. 86 (1957) 28–56.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) 6–19.Google Scholar

7. Zemanian, A. H., Distribution theory and transform analysis. (McGraw-Hill, New York, 1965).Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

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Haruki, Hiroshi 1970. On the Functional Equationf(x+t,y)+f(x−t,y)=f(x,y+t)+f(x, y−t). Aequationes Mathematicae, Vol. 5, Issue. 1, p. 118.

von Leissner, W. 1971. Achte Internationale Tagung über Funktionalgleichungen in Oberwolfach vom 2.–8. August 1970. aequationes mathematicae, Vol. 6, Issue. 1, p. 90.

McKiernan, M. A. 1972. The general solution of some finite difference equations analogous to the wave equation. Aequationes Mathematicae, Vol. 8, Issue. 3, p. 263.

Flemming, D. P. 1972. Generalizations of the Wave Equation which Allow Weak and Strong Discontinuities. SIAM Journal on Applied Mathematics, Vol. 23, Issue. 2, p. 221.

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Haruki, Shigeru 1980. Note on the equation $$\left( {\mathop {\Delta ^2 }\limits_{x,t} - \mathop {\Delta ^2 }\limits_{y,t} } \right)f = 0$$. Aequationes Mathematicae, Vol. 21, Issue. 1, p. 49.

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Chlebík, Miroslav and Král, Josef 1994. Removability of the wave singularities in the plane. Aequationes Mathematicae, Vol. 47, Issue. 2-3, p. 123.

Chung, Jukang Kannappan, Palaniappan and Sahoo, Prasanna K. 1997. On trigonometric functional equations of rectangular type. Aequationes Mathematicae, Vol. 53, Issue. 1-2, p. 36.

Fenyö, I. 2010. Functional Equations and Inequalities. p. 43.

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An Analogue of the Wave Equation and Certain Related Functional Equations

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