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The extremal truncated moment problem. (English) Zbl 1145.47012

Integral Equations Oper. Theory 60, No. 2, 177-200 (2008); addendum ibid. 61, No. 1, 147-148 (2008).
If \(\mu\) is a \(d\)-dimensional positive Borel measure, then its moments are \(\beta_i=\int x^i\,d\mu\), where \(x^i=\prod_{k=1}^d x_k^{i_k}\). The truncated moment problem is to find a representing \(\mu\), given the moments \(\{\beta_i : i\in{\mathbb Z}_+^d, | i| \leq 2n\}\). These moments define a moment functional \(\Lambda\) on the space of polynomials \({\mathcal P}_n\) of degree \(\leq n\). The moment matrix \({\mathcal M}(n)\) containing all these moments is positive semidefinite (it is Hankel when \(d=1\)). Its kernel are the vectors whose entries are the coefficients of the polynomials \(p\in K=\{p\in{\mathcal P}_n:\Lambda(p)=0\}\). Finally, define the variety \({\mathcal V}=\{z\in{\mathbb C}: p(z)=0, \forall p\in K\}\). If a solution to the moment problem exists, then its support can only be part of \({\mathcal V}\). Necessary conditions for the solvability of the truncated moment problem are that (1) \({\mathcal M}(n)\geq0\), (2) if \(p,q,pq\in{\mathcal P}_n\) and \(p\in K\), then \(pq\in K\), (3) \(r=\text{rank}\;{\mathcal M}(n)\leq \text{card}\;{\mathcal V}=v\).
In an attempt to answer the question if these conditions are also sufficient, it is proved that for \(d=2\), \(n=3\), \(r=v=7\) or 8, and for a particular moment matrix, the answer is positive. If (2) is replaced by the stronger condition that all \(p\in{\mathcal P}_{n}\) vanishing on \({\mathcal V}\) belong to \(K\), the existence of a representing \(r\)-atomic \(\mu\) is proved in the case \(r=v\). Some positive evidence is given that this is also suffcient when \(r\leq v\), but no hard proof is given yet.

MSC:

47A57 Linear operator methods in interpolation, moment and extension problems
44A60 Moment problems
42A70 Trigonometric moment problems in one variable harmonic analysis
30E05 Moment problems and interpolation problems in the complex plane
15B57 Hermitian, skew-Hermitian, and related matrices
15-04 Software, source code, etc. for problems pertaining to linear algebra
47N40 Applications of operator theory in numerical analysis
47A20 Dilations, extensions, compressions of linear operators
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