Truncated moment problems on positive-dimensional algebraic varieties

Abstract

This manuscript transfers the main aspects of Prony's method from finitely-supported measures to the classes of signed or non-negative measures supported on algebraic varieties of any dimension. In particular, we show that the Zariski closure of the support of these measures is determined by finitely many moments and can be computed from the kernel of certain moment matrices.

Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

“…[22]. Similarly, it holds for sufficiently large n if µ ∈ M + ; see for example [25,Theorem 2.10] or [47,Proposition 4.10].…”

“…In this case, it is possible to rephrase the hypothesis in terms of a non-square moment matrix of suitable size (cf. [47,Theorem 4.3]) to obtain a statement similar to Theorem 3.1.…”

“…If µ ∈ M is not finitely-supported, then V (ker T n ) = V can fail to hold for any n ∈ N (cf. [47,Example 4.9]). In this case, it is possible to rephrase the hypothesis in terms of a non-square moment matrix of suitable size (cf.…”

“…The positivedimensional case on the other hand is more involved. Specific curves in a two-dimensional domain are identified by the kernel of moment matrices in [34,13,33], more general discussions can be found in [25] and [47]. In another line of work, Christoffel functions offer interesting guarantees both in terms of support identification [24] or approximation on the support [20,28,35], but, to the best of our knowledge, require strong regularity assumptions, and only come with separate guarantees on and outside the support of the measure.…”