An analogue of the Riesz–Haviland theorem for the truncated moment problem

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• 作者：Raú ; l E. Curto ; Lawrence A. Fialkow
• 关键词：Truncated moment problem ; K-moment problems ; Riesz functional ; Riesz– ; Haviland theorem ; Moment matrix extension ; Flat extensions of positive matrices ; Semialgebraic sets ; Localizing matrices ; Positive functional
• 刊名：Journal of Functional Analysis
• 出版年：2008
• 出版时间：15 November 2008
• 年：2008
• 卷：255
• 期：10
• 页码：2709-2731
• 全文大小：278 K

文摘

Let b2;≡b2;⁽²ⁿ⁾={b2;b>ib>}b>i2nb> denote a d-dimensional real multisequence, let K denote a closed subset of , and let . Corresponding to b2;, the Riesz functional is defined by L(∑ab>ib>xⁱ):=∑ab>ib>b2;b>ib>. We say that L is K-positive if whenever and pb>Kb>0, then L(p)0. We prove that b2; admits a K-representing measure if and only if Lb>b2;b> admits a K-positive linear extension . This provides a generalization (from the full moment problem to the truncated moment problem) of the Riesz–Haviland theorem. We also show that a semialgebraic set solves the truncated moment problem in terms of natural “degree-bounded” positivity conditions if and only if each polynomial strictly positive on that set admits a degree-bounded weighted sum-of-squares representation.