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The Moment Problem on Compact Semi-Algebraic Sets

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The Moment Problem

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 277))

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Abstract

In this chapter we begin the study of the multidimensional moment problem. The passage to dimensions d ≥ 2 brings new difficulties and unexpected phenomena. In Sect. 3.2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions. It seems to be natural to look for similar characterizations in higher dimensions as well. This leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi-algebraic sets. In this chapter we treat this approach for basic closed compact semi-algebraic subsets of \(\mathbb{R}^{d}\). It turns out that for such sets there is a close interaction between the moment problem and Positivstellensätze for strictly positive polynomials.

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  1. Mathematisches Institut, Universität Leipzig, Leipzig, Germany

    Konrad Schmüdgen

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  1. Konrad Schmüdgen
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Schmüdgen, K. (2017). The Moment Problem on Compact Semi-Algebraic Sets. In: The Moment Problem. Graduate Texts in Mathematics, vol 277. Springer, Cham. https://doi.org/10.1007/978-3-319-64546-9_12

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