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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’ equationsf(x + y - xy) +f(xy) =f(x) +f (y ),by Neagu [4]for the Pompeiu equationf(x+y+xy)=f(x)+f(y)+f(x)f(y)and by Swiatak [6].

Abstract.The Pompeiu functional equation is defined by Neagu for Schwartz distributions. His method is extended to the four Cauchy functional equations by means of two new operators Q* and R* on 3'(I).The Cauchy equations in distributions reduce to the classical equations when the solutions are regular distributions, i.e. locally integrable functions.

Powers and products of distributions have not as yet been defined to hold true in general. In this paper, we choose a fixed 6-sequence and use the concept of neutrix limit to give meaning to the distributions Sk and (~3 ' )~ for some k. These may be regarded as powers of DIRAC delta functions.