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The stability of Cauchy equations in the space of Schwartz distributions. (English) Zbl 1053.39043

The authors reformulate the classical result of D.H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] concerning the stability of the Cauchy functional equation f(x+y)=f(x)+f(y). Let S=S( n ) denotes the Schwartz space of rapidly decreasing functions on n and S ' (its dual) the space of tempered distributions. For uS ' the pullbacks uA, uP 1 , uP 2 of u are defined by

uA,ϕ(x+y)=u,ϕ(x-y,y)dy;
uP 1 ,ϕ(x+y)=u,ϕ(x,y)dy;
uP 2 ,ϕ(x+y)=u,ϕ(x,y)dx

for all test functions ϕS( 2n ). Finally, for vS ' , vε means that |v,ϕ|εϕ L 1 for ϕS. A distribution uS ' is called ε-additive if

uA-uP 1 -uP 2 ε·

The main result of the paper states that each ε-additive distribution u can be written uniquely in the form u=c·x+h(x) with c n and a bounded measurable function h such that h L ε. Similar investigations are conducted for a counterpart of the Cauchy equation f(x+y)=f(x)f(y) and its superstability.


MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
46F10Operations with distributions (generalized functions)