Extremal sextic truncated moment problems

Inverse problems naturally occur in many branches of science and mathematics. An inverse problem entails finding the values of one or more parameters using the values obtained from observed data. A typical example of an inverse problem is the inversion of the Radon transform. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This problem is intimately connected with image reconstruction for X-ray computerized tomography.

Moment problems are a special class of inverse problems. While the classical theory of moments dates back to the beginning of the 20th century, the systematic study of truncated moment problems began only a few years ago. In this dissertation we will first survey the elementary theory of truncated moment problems, and then focus on those problems with cubic column relations.

For a degree 2n real d-dimensional multisequence β ≡ β ⁽²ⁿ⁾ ={β _i}_{i∈Z^d+,|i|≤2n} to have a representing measure μ, it is necessary for the associated moment matrix Μ(n) to be positive semidefinite, and for the algebraic variety associated to β, V_β, to satisfy rank Μ(n)≤ card V_β as well as the following consistency condition: if a polynomial p(x)≡ ∑_|i|≤2na_ixⁱ vanishes on V_β, then Λ(p):=∑_|i|≤2na_iβ_i=0. In 2005, Professor Raúl Curto collaborated with L. Fialkow and M. Möller to prove that for the extremal case (Μ(n)= V_β), positivity and consistency are sufficient for the existence of a (unique, rank Μ(n)-atomic) representing measure.

In joint work with Professor Raúl Curto we have considered cubic column relations in M(3) of the form (in complex notation) Z³=itZ+ubar Z, where u and t are real numbers. For (u,t) in the interior of a real cone, we prove that the algebraic variety V_β consists of exactly 7 points, and we then apply the above mentioned solution of the extremal moment problem to obtain a necessary and sufficient condition for the existence of a representing measure. This requires a new representation theorem for sextic polynomials in Z and bar Z which vanish in the 7-point set V_β. Our proof of this representation theorem relies on two successive applications of the Fundamental Theorem of Linear Algebra. Finally, we use the Division Algorithm from algebraic geometry to extend this result to other situations involving cubic column relations.