Vol. 68, No. 1, 1977

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Moments of measures on convex bodies

P. H. Maserick

Vol. 68 (1977), No. 1, 135–152

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) = Kt1n1t2n2t3n3 (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.

Mathematical Subject Classification 2000
Primary: 44A50, 44A50
Secondary: 46J99, 22A20
Received: 7 October 1975
Published: 1 January 1977
P. H. Maserick