The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

Dio, Philipp J. di

doi:10.1090/proc/14499

Cited by 6 publications

(2 citation statements)

References 43 publications

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“…Since X is a topological space which consists of at most H F V (2d) − 1 pathconnected components, 1 ∈ R[X ], and R[X ] consists of continuous functions we have that s A (X ) consists of at most H F V (2d) − 1 path-connected components. All conditions of [13,Thm. 12] are fulfilled which implies the upper bound.…”

Section: Proposition 41 Every Moment Functional L : R[x ] ≤2d → R Is a Conic Combination Of At Most H F V (2d) Point Evaluations L X I Wimentioning

confidence: 99%

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Dio

Kummer

2021

Math. Ann.

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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, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

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Section: Proposition 41 Every Moment Functional L : R[x ] ≤2d → R Is a Conic Combination Of At Most H F V (2d) Point Evaluations L X I Wimentioning

confidence: 99%

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Dio

Kummer

2021

Math. Ann.

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“…In this section we aim to provide the following generalization of the celebrated theorem of Tchakaloff [78, Theorem II]. On the other hand, it was proved in [4] (see also [68], [71,Corollary 1.25] and [24,Theorem 6]) that a soluble B-truncated moment problem, with B finite dimensional linear subspace of C(K) and K locally compact, always admits a finitely atomic representing measure.…”

Section: Support Of the Representing Measuresmentioning

confidence: 99%

The Truncated Moment Problem for Unital Commutative R-Algebras

Curto,

Ghasemi,

Infusino

et al. 2020

Preprint

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Transformations of Moment Functionals

Dio

2022

Integr. Equ. Oper. Theory

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In measure theory several results are known how measure spaces are transformed into each other. But since moment functionals are represented by a measure we investigate in this study the effects and implications of these measure transformations to moment funcationals, especially with dimensionality reduction. We gain characterizations of moment functionals. Among other things we show that for a compact and path connected set $$K\subset \mathbb {R}^n$$ K ⊂ R n there exists a measurable function $$g:K\rightarrow [0,1]$$ g : K → [ 0 , 1 ] such that any linear functional $$L:\mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] → R is a K-moment functional if and only if it has a continuous extension to some $$\overline{L}:\mathbb {R}[x_1,\dots ,x_n]+\mathbb {R}[g]\rightarrow \mathbb {R}$$ L ¯ : R [ x 1 , ⋯ , x n ] + R [ g ] → R such that $$\tilde{L}:\mathbb {R}[t]\rightarrow \mathbb {R}$$ L ~ : R [ t ] → R defined by $$\tilde{L}(t^d):= \overline{L}(g^d)$$ L ~ ( t d ) : = L ¯ ( g d ) for all $$d\in \mathbb {N}_0$$ d ∈ N 0 is a [0, 1]-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function $$f:[0,1]\rightarrow K$$ f : [ 0 , 1 ] → K independent on L such that the representing measure $$\tilde{\mu }$$ μ ~ of $$\tilde{L}$$ L ~ provides the representing measure $$\tilde{\mu }\circ f^{-1}$$ μ ~ ∘ f - 1 of L. We also show that every moment functional $$L:\mathcal {V}\rightarrow \mathbb {R}$$ L : V → R is represented by $$\lambda \circ f^{-1}$$ λ ∘ f - 1 for some measurable function $$f:[0,1]\rightarrow \mathbb {R}^n$$ f : [ 0 , 1 ] → R n where $$\lambda $$ λ is the Lebesgue measure on [0, 1].

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The multidimensional truncated moment problem: Gaussian and log-normal mixtures, their Carathéodory numbers, and set of atoms

Cited by 6 publications

References 43 publications

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

The Truncated Moment Problem for Unital Commutative R-Algebras

Transformations of Moment Functionals

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