This paper is concerned with
the representation, in terms of convolutions with pseudomeasures, of continuous
linear operators which commute with translations and which transform continuous
functions with compact supports on a Hausdorff locally compact Abelian group G
into restricted types of Radon measures on G. The two main theorems each assert
that any such operator T is of the form Tf = s ∗ f for a suitably chosen
pseudomeasure s on G; the assertions differ in detail in respect of the hypotheses
imposed on the range of T. The second theorem is an extension of Proposition 2 of [1]
from the case in which G is a finite product of lines and/or circles to the general
situation.
