On a minimal solution for the indefinite multidimensional truncated moment problem

Abstract

We will consider the indefinite multidimensional truncated moment problem. Necessary and sufficient conditions for a given truncated multisequence to have a signed representing measure with minimal cardinality of its support are given by the existence of a rank preserving extension of a multivariate Hankel matrix (built from the given truncated multisequence) such that the corresponding associated polynomial ideal is real radical. This result is a special case of a more general characterisation of truncated multisequences with a minimal complex representing measure whose support is symmetric with respect to complex conjugation (which we will call quasi-complex). One motivation for our results is the fact that positive semidefinite truncated multisequences need not have a positive representing measure. Thus, our main result gives the potential for computing a signed representing measure μ with Jordan decomposition μ=μ+−μ−, where cardμ− is small. We will illustrate this point on concrete examples.