Geodesic
The Pexider Functional Equations in Distributions
Canadian journal of mathematics, Tome 42 (1990) no. 2, pp. 304-314

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The Cauchy functional equations have been studied recently for Schwartz distributions by Koh in [3]. When the solutions are locally integrate functions, the equations reduce to the classical Cauchy equations (see [1]):(1) f(x+y)=f{x)+f(y)(2) f(x+y)=f(x)f(y)(3) f(xy)=f(x)+f(y)(4) f(xy)=f(x)f(y).Earlier efforts to study functional equations in distributions were given by Fenyö [2]for the Hosszu’ equations f(x + y - xy) +f(xy) =f(x) +f (y ),by Neagu [4]for the Pompeiu equation f(x+y+xy)=f(x)+f(y)+f(x)f(y)and by Swiatak [6]. 
Deeba, E. Y.; Koh, E. L. The Pexider Functional Equations in Distributions. Canadian journal of mathematics, Tome 42 (1990) no. 2, pp. 304-314. doi: 10.4153/CJM-1990-017-6
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[1] 1. Aczel, J., Lectures on functional equations and their applications (Academic Press, New York, 1966). Google Scholar

[2] 2. Fenyö, I., On the general solution of a functional equation in the domain of distributions, Aequationes Math. 3 (1969), 236–246. Google Scholar

[3] 3. Koh, E.L., The Cauchy functional equations in distributions, Proc. Amer. Math. Soc. 106 (1989), 641–647. Google Scholar

[4] 4. Neagu, M., About the Pompeiu equation in distributions, Inst. Politehn. “Traran Vaia” Timis. Sem. Mat. Fiz. (1984), 62–66. Google Scholar

[5] 5. Schwartz, L., Theorie des distributions (Hermann, Paris, 1966). Google Scholar

[6] 6. Swiatak, H., On the regularity of the distributional and continuous solutions of the functional equations Aequationes Math. 1 (1968), 6–19. Google Scholar

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