The subject of this paper is the use of the theory of Schwartz distributions and approximate identities in studying the functional equation
The aj
’s and b are complex-valued functions defined on a neighbourhood, U, of 0 in R
m
, hj
. U → R
n
with hj
(0) = 0 and fj
, g: R
n
→ C for 1 ≦ j ≦ N. In most of what follows the aj
's and hj
's are assumed smooth and may be thought of as given. The fj
‘s, b and g may be thought of as the unknowns. Typically we are concerned with locally integrable functions f
1, … , fN
such that, for each s in U, (1) holds for a.e. (almost every) x ∈ R
n
, in the sense of Lebesgue measure.