In this paper the problem of
moments is viewed as one of identifying a class of functions on a semigroup with a
class of measures. We present integral representation theorems for linear functionals
on algebras (Theorems 2.1 and 2.2) which enable us to solve moment problems for a
wide class of compact sets. In particular if K is any compact convex subset of
R3-with nonvoid interior, necessary and sufficient conditions are given for a triple
indexed sequence f(n1,n2,n3) to admit an integral representatlon of the form
f(n1,n2,n3) =∫Kt1n1t2n2t3n3dμ(t)(t = (t1,t2,t3)). Here, of course the semigroup S
considered is all triples of nonnegative integers under coordinate addition. As in the
case of Hausdorff’s “little moment problems” the solution depends on certain linear
combinations of shift operators.
