The Cauchy functional equations in distributions

Abstract

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.

d’Alembert functional equations in distributions

“…In order to represent the left-hand side of equation (1.2), one needs to define two additional operators.This will be discussed in the next section. Moreover, we remark that the operator (2+ is similar to the operator Q defined and studied in [8,12]. The adjoints of these operators are Q* and Q*_ and are defined from 2'{I) into 2'il2) by…”

“…The proof of these properties is analogous to [12] and thus we shall omit it. In order to define equation ( 1.2) in distributions one needs to define a tensor product operator on 2'{I) x 3'{I)…”

“…Since the action of the partial differential operators Dx and D2 on Q+ and Q-differs slightly from results obtained in [8,12], we shall include the proof of the following proposition, which is also essential in the solution of the distributional analog of equation (1.2). …”

Reformulation of some functional equations in the space of Gevrey distributions and regularity of solutions

“…Many well-known functional equations have been studied in the Schwartz distributions (see [2], [3], [4], [5], [10] and [11] etc.). Actually it was shown that most of these functional equations in the distributions reduce to the classical ones.…”

On generalizations of the Pompeiu functional equation

“…This functional equation is known as the Pompeiu .furtc.tiortal equation [3,4]. To see this, add 1 to both sides of (PE) and write F(x) 1 + f(x).…”