Abstract
In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, mathbb {R}^n, and [0,1]^n. We also treat moment problems with small gaps. We find that for every varepsilon >0 and din mathbb {N} there is a nin mathbb {N} such that we can construct a moment functional L:mathbb {R}[x_1,cdots ,x_n]_{le d}rightarrow mathbb {R} which needs at least (1-varepsilon )cdot left( {begin{matrix} n+d nend{matrix}}right) atoms l_{x_i}. Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals L:mathbb {R}[x_1,cdots ,x_n]_{le 2d}rightarrow mathbb {R} which need to be extended to the worst case degree 4d, tilde{L}:mathbb {R}[x_1,cdots ,x_n]_{le 4d}rightarrow mathbb {R}, in order to have a flat extension.
Highlights
The theory of moment sequences is a field of diverse applications and connections to numerous other mathematical fields, see e.g. [1,22,29,30,31,33,34,36,38, 48,50,52], and references therein
A crucial fact in the theory of truncated moment sequences is the Richter (Richter– Rogosinski–Rosenbloom) Theorem [43,44,45] which states that every truncated moment sequence is a convex combination of finitely many Dirac measures, see Theorem 2.2
We treat moment sequences with small gaps, moment sequences of measures supported on algebraic varieties (Sect. 4), and the multidimensional polynomial case on Rn and [0, 1]n
Summary
The theory of (truncated) moment sequences is a field of diverse applications and connections to numerous other mathematical fields, see e.g. [1,22,29,30,31,33,34,36,38, 48,50,52], and references therein. The Carathéodory number is the minimal number N such that every truncated moment sequence (with fixed truncation) is a sum of N atoms, i.e., Dirac measures. It has been studied in several contexts but in most cases the precise value of the Carathéodory number is not known [15,16,32,39,42,43,46,53]. 3), moment sequences of measures supported on algebraic varieties For moment functionals with small gaps we find explicit lower and upper bounds for dimension n = 1 based on Descartes’ rule of signs, see Theorem 3.7. For literature on flat extensions in this context see [8,9,36,48] and the references therein
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