The Pexider Functional Equations in Distributions

Abstract

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].

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 this proposition can be carried out exactly as in [8]. The product operator F will be used, therefore, to represent the product f{x)g{y) of equation (1.2) when the equation is interpreted in distributions.…”

“…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.…”

“…Actually, in this paper we reformulate the following classical functional equations in Gevrey distributions: (1) f (x + y) = f(x) + f(y), (2) f (x + y) = f(x) · f(y),…”