The truncated moment problem asks to characterize linear functionals over the space of polynomials
of a given degree, which can be represented as integration over the positive Borel measure $\mu$ with
support on a given closed subset of $\real^n$. We can solve this by observing the properties of the
corresponding moment matrix $\mathcal{M}$. In this work we are going to study the cases that have two variables. We then label the columns of $\mathcal{M}$ with monomials $x^i y^j$. In this way, every element of the kernel of $\mathcal{M}$ can be expressed as a symbolic zero set of some polynomial. In our approach we will
assume that $\mathcal{M}$ is singular and in this way we will get rid of one of the variables. Afterwards
we will solve the corresponding one dimensional problem. We are going to crucially rely on some results from
graph theory, since there are certain moments in the sequence that are missing.
|
