Application of Localization to the Multivariate Moment Problem Ii

The paper is a sequel to the paper [5] by the same author. A new criterion is presented for a PSD linear map L : R[x] → R to correspond to a positive Borel measure on Rn. The criterion is stronger than Nussbaum’s criterion in [6] and is similar in nature to Schmüdgen’s criterion in [5] [7]. It is also explained how the criterion allows one to understand the support of the associated measure in terms of the non-negativity of L on a quadratic module of R[x]. This latter result extends a result of Lasserre in [3]. The techniques employed are the same localization techniques employed already in [4] and [5], specifically one works in the localization of R[x] at p = ∏n i=1(1 + x 2 i ) or p′ = ∏n−1 i=1 (1 + x 2 i ). This paper is a sequel to the earlier paper [5]. We present a couple of interesting and illuminating results which were inadvertently overlooked when [5] was written; see Theorems 0.1 and 0.5 below. Theorem 0.1 extends an old result of Nussbaum in [6]. See Theorem 0.3 below for a statement of Nussbaum’s result. The density condition (0.1) appearing in Theorem 0.1 is weaker than the Carleman condition (0.2) appearing in Nussbaum’s result. Theorem 0.5 shows how condition (0.1) allows one to read off information about the support of the measure from the nonnegativity of the linear functional on a quadratic module. This illustrates how natural condition (0.1) is. Theorem 0.5 extends a result of Lasserre in [3]. We recall some terminology and notation from [4] and [5]. For an R-algebra A (commutative with 1), a quadratic module of A is a subset M of A such that 1 ∈ M , M + M ⊆ M and fM ⊆ M for all f ∈ A. ∑ A denotes the set of all (finite) sums of squares of A. ∑ A is the unique smallest quadratic module of A. A linear map L : A → R is said to be PSD (positive semidefinite) if L(f) ≥ 0 for all f ∈ A, equivalently, if L( ∑ A) ⊆ [0,∞). Define R[x] := R[x1, . . . , xn], C[x] := C[x1, . . . , xn]. If μ is a positive Borel measure on R having finite moments, i.e., ∫ xdμ is well-defined and finite for all monomials x := x1 1 . . . x kn n , kj ≥ 0, j = 1, . . . , n, the PSD linear map Lμ : R[x] → R is defined by Lμ(f) = ∫ fdμ. If 2000 Mathematics Subject Classification. Primary 44A60 Secondary 14P99.