一类差分的刻画

Characterizations for a Kind of Difference

函数 Δ:G×G→H是 Cauchy差分 ,即存在函数 f:G→H使 Δ( x,y) =f( x+ y) - f( x) - f( y) ,其中 G和 H是 abelian群并且 H可除 ,推导出使上式成立的充分必要条件是 Δ( x,y) =Δ( y,x)和Δ( x,y) +Δ( x+ y,z) =Δ( x,y+ z) +Δ( y,z)。推广了 Cauchy差分的形式并且利用函数方程组给出了函数 F:G× G→ H具有差分表示 F( x,y) =f ( x+ y) + f ( x- y) - nf ( x) - nf ( y)的几种刻画 ,其中 n是一正整数 ,f是 G到 H的函数。

The function has the Cauchy difference with G and H being abelian groups and H divisible, i.e. there exists a function f:G→H that makes Δ (x,y)=f(x+y)-f(x)-f(y) if and only if Δ (x,y) =Δ( y,x ) and Δ( x,y )+Δ( x+y,z )=Δ( x,y+z )+Δ( y,z ). The form of the Cauchy difference is generalized, and several characterizations are given for the function F:G×G→H with the difference representation F(x,y)=f(x+y)+f(x-y)-nf(x)-nf(y) , by means of systems of functional equations, where n is a positive integer and f is a function from G into H .