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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 $\left(*\right)$ ${\sum }_{j=0}^{m}{F}_{j}\left({a}_{j}x+{b}_{j}y\right)={\sum }_{k=1}^{n}{G}_{k}\left(x\right){H}_{k}\left(y\right)$. 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 $\left(*\right)$ $\left({D}_{x}$ and ${D}_{y}$ are the derivations with respect to $x$ or $y$, respectively), one gets a similar equation with (by 1) lower $m$.

##### MSC:
 39B32 Functional equations for complex functions 39B42 Matrix and operator functional equations 46F10 Operations with distributions (generalized functions) 28A35 Measures 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 $H\left(x\circ y\right)={{\Sigma }}_{i}{f}_{i}\left(x\right){g}_{i}\left(y\right)$ , 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.