Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems

By L. Fialkow, Jiawang Nie. 2009

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Abstract We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence y , we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set K ⊆ R n if and only if the associated Riesz functional L y is K -positive. For a determining set K , we prove that if L y is strictly K -positive, then y admits a representing measure supported in K . As a consequence, we are able to solve the truncated K -moment problem of degree k in the cases: (i) ( n , k ) = ( 2 , 4 ) and K = R 2 ; (ii) n ⩾ 1 , k = 2 , and K is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.

Subfields: Applied Mathematics, Statistics and Probability

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