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On a functional equation of Aczél and Chung. (English) Zbl 0791.39008
The equation in the title, the locally integrable solutions of which have been determined in [Studia Sci. Math. Hungar. 17, 51-67 (1982; Zbl 0553.39005)] is (*) j=0 m F j (a j x+b j y)= k=1 n G k (x)H k (y). The author solves the analogue of this equation for (Laurent Schwartz) distributions by first reducing it to the case where m=1. This can be done because distributions are differentiable and, by applying b p D x -a p D y to (*) (D x and D y are the derivations with respect to x or y, respectively), one gets a similar equation with (by 1) lower m.

MSC:
39B32Functional equations for complex functions
39B42Matrix and operator functional equations
46F10Operations with distributions (generalized functions)
28A35Measures and integrals in product spaces
References:
[1]Aczél, J. andChung, J. K.,Integrable Solutions of Functional Equations of a General Type, Studia Sci. Math. Hungar.17 (1982), 51–67.
[2]Aczél, J. andDhombres, J.,Functional equations in several variables, Cambridge University Press, 1989.
[3]Deeba, E. Y. andKoh, E. L.,Coupled Functional Equations in Distributions, preprint.
[4]Gel’fand, I. M. andShilov, G. E.,Generalized Functions, Vol. I, Academic Press, New York and London, 1964.
[5]Hörmander, Lars,The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
[6]Jarai, A.,A remark to a paper of J. Aczél and J. K. Chung, Studia Sci. Math. Hungar.19 (1984), 273–274.
[7]McKiernan, M. A., Equations of the form H(xy)=Σ i f i (x)g i (y) , Aequationes Math.16 (1977), 51–58. · Zbl 0392.39004 · doi:10.1007/BF01836418
[8]Rudin, Walter,Functional Analysis, McGraw-Hill, New York, 1973.
[9]Székelyhidi, L.,On the Levi-Civita Functional Equation, Berichte, Nr. 301, der Mathematisch–Statistischen Sektion in der Forschungsgesellschaft Joanneum–Graz, 1988.
[10]Vincze, E.,Eine allgemeinere Methode in der Theorie der Funktionalgleichungen I, Publ. Math. Debrecen9 (1962), 149–163.