The Truncated Moment Problem for Unital Commutative R-Algebras

Abstract

We investigate when a linear functional $L$ defined on a linear subspace $B$ of a unital commutative real algebra $A$ admits an integral representation w.r.t. a positive Radon measure supported on a closed subset $K$ of the character space of $A$. We provide a criterion for the existence of such a representation for $L$ when $A$ is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for $A$ not necessarily equipped with a topology. This allows us to extend well-known classical results on truncated moment problems.

Figures (9)

• Figure 1: On the left, diagram of $B_0$ (dots) and $B_1$ (squares); on the right, diagram of $B_N$ (diamonds)
• Figure 2: Diagrams of monomial powers in the Classical Truncated Moment Problem (left) and the Rectangular Truncated Moment Problem (right)
• Figure 3: Diagrams of monomials for $\mathcal{C}=\{1,s,st\}$
• Figure 4: Monomial diagram for $\mathcal{C}=\{1,s,t,st,t^2,s^2t,s^3t,\cdots\}$
• Figure 5: Diagrams of monomial powers in the Classical Truncated Moment Problem (left, circles) and the Rectangular Truncated Moment Problem (right, circles), and for their respective flat extensions (circles and squares)

Theorems & Definitions (56)

• Remark 1.1
• Definition 1.2
• Proposition 1.3
• proof
• Proposition 1.4
• proof
• Remark 1.5
• Definition 2.1
• Theorem 2.2
• proof