Abstract
Let \({\mathbf{K} \subset \mathbb{R}^n}\) be a compact basic semi-algebraic set. We provide a necessary and sufficient condition (with no à priori bounding parameter) for a real sequence y = (y α), \({\alpha \in \mathbb{N}^n}\) , to have a finite representing Borel measure absolutely continuous w.r.t. the Lebesgue measure on K, and with a density in \({\cap_{p \geq 1} L_p(\mathbf{K})}\) . With an additional condition involving a bounding parameter, the condition is necessary and sufficient for the existence of a density in L ∞(K). Moreover, nonexistence of such a density can be detected by solving finitely many of a hierarchy of semidefinite programs. In particular, if the semidefinite program at step d of the hierarchy has no solution, then the sequence cannot have a representing measure on K with a density in L p (K) for any p ≥ 2d.
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Lasserre, J.B. Borel measures with a density on a compact semi-algebraic set. Arch. Math. 101, 361–371 (2013). https://doi.org/10.1007/s00013-013-0557-5
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DOI: https://doi.org/10.1007/s00013-013-0557-5