Solving the real cubic truncated moment problem in the singular case

Abstract

In the most intuitive way, the problem of moments seeks to represent the terms of a given sequence using an integral. So, it is about determining a measure that allows this representation. In mathematical analysis, the problem of moments (MP) occupies an important place in the work of many researchers. Several generalizations and extensions of the original version, attributed to T. J. Stieltjes, have emerged. It has been a source of inspiration for many developments in many branches of mathematics as well as in other fields. In this article, we are interested in a class of two-dimensional MP. Precisely, we deal with the problem of the cubic real truncated moment in the case where the associated moment matrix is singular. The main target is to provide a complete solution via a minimal atomic-representative measure. This was possible by the use of an algorithmic method based on the construction of some flat extension of the associated moment matrix. The implementation of this approach is illustrated by some numerical examples using Mathematica software.