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].