Abstract
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.
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