Algebraic–exponential Data Recovery from Moments

Abstract

Let $$\mathbf {G}$$G be a bounded open subset of Euclidean space whose boundary $$\Gamma $$Γ is algebraic, i.e., contained in the real zero set of finitely many polynomials. Under the assumption that the degree d of this variety is given, and the power moments of the Lebesgue measure on $$\mathbf {G}$$G are known up to order 3d, we describe an algorithmic procedure for obtaining a polynomial vanishing on $$\Gamma $$Γ. The particular case of semi-algebraic sets defined by a single polynomial inequality raises an intriguing question related to the finite determinateness of the full moment sequence. The more general case of a measure with density equal to the exponential of a polynomial is treated in parallel. Our approach relies on Stokes' Theorem on spaces with singularities and simple Hankel-type matrix identities.