The truncated moment problem via homogenization and flat extensions

This paper studies how to solve the truncated moment problem (TMP) via homogenization and flat extensions of moment matrices. We first transform TMP to a homogeneous TMP (HTMP), and then use semidefinite programming (SDP) techniques to solve HTMP. Our main results are: (1) a truncated moment sequence (tms) is the limit of a sequence of tms admitting measures on Rn if and only if its homogenized tms (htms) admits a measure supported on the unit sphere in Rn+1; (2) an htms admits a measure if and only if the optimal values of a sequence of SDP problems are nonnegative; (3) under some conditions that are almost necessary and sufficient, by solving these SDP problems, a representing measure for an htms can be explicitly constructed if one exists.