Moment Problems for Compact Sets

The solvability of the Hausdorff moment problem for an arbitrary compact subset of Euclidean n-space is shown to be equivalent to the nonnegativity of a family of quadratic forms derived from the given moment sequence and the given compact set. A variant theorem for the one-dimensional case and an analogous theorem for the trigonometric moment problem are also given. The one-dimensional theorems are similar to theorems of Kreïn and Nudel'man [11], but the proofs, unlike those in [11], do not depend on the existence of a standard form for polynomials which are nonnegative on a given compact set. 1. Let n be a positive integer, R the real line, and JV the set of all nonnegative integers. Let Rn be Euclidean n-space, i.e. the set of all n-tuples x = (xi, x2,..., xn) of real numbers; and let JVn be the set of all n-tuples j — (ji,j2, ■ ■. ,jn) of nonnegative integers. The elements of Nn will be called multi-indices. We shall often write xJ for the product _Jl rJl . . . _7„ •Cl •c2 xn where x G Rn and j G Nn. In this section we shall solve the following n-dimensional generalization of the Hausdorff moment problem [9, 11]: given a compact subset K of Rn and a multiply indexed sequence of real numbers {cj : j e Nn} = {c(ji,j2, ...,jn): (J1J2, . •., jn) G Nn}, find necessary and sufficient conditions that there exist a positive Borel measure p, supported on K, such that Cj = jK xJ dp(x), i.e. c(ji,J2,---,jn)= / xilx32 ■■■xn"dp(x), Jk for every multi-index j; p is supported on K if p(E) — p(E n K) for every Borel subset E of Rn. Our solution will be that certain quadratic forms are nonnegative; the quadratic forms are determined by the "moment sequence" {c}•: j G Nn} and by a decomposition of the compact set K. In order to state our solution, we must first describe the decomposition of K, which is not unique. Let [a,b] = [cn,61] x [a2,b2] x ■ x [an,bn] be a rectangle containing K, where ai < bi, a2 < b2,...,an < bn, and a — (ai,a2,... ,an), b = (bi,b2,...,bn)Since the topology of Rn has a countable Received by the editors August 12, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 44A60, 42A70; Secondary 47B15. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page