The Classical Moment Problem and Some Related Questions in Analysis

Akhiezer, N. I.; Kemmer, N.

doi:10.1137/1.9781611976397

Cited by 507 publications

(425 citation statements)

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“…The connection between the truncated moment problem and non-negative polynomials run deep and can be found in a large number of publications on the truncated moment problem, see e.g. [1,31,34,36,38,48].…”

Section: Lemma 61 Let a Be A Finite Dimensional Vector Space Of Measurable Functions With Basismentioning

confidence: 99%

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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: Lemma 61 Let a Be A Finite Dimensional Vector Space Of Measurable Functions With Basismentioning

confidence: 99%

“…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. For more on recent advances in the reconstruction of measures from moments see e.g.…”

Section: Introductionmentioning

confidence: 99%

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Dio

Kummer

2021

Math. Ann.

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“…Το πρόβλημα της μοναδικότητας του μέτρου μ(χ) είναι επίσης ισοδύναμο με το πρό βλημα: Πότε ο συμμετρικός τελεστής Τ είναι αυτοσυζυγής, ή επιδέχεται μοναδική αυτοσυζυγή επέκταση, [ 105 ], [ 99 ], [ 2 ]. Σ' αυτήν την περίπτωση το φάσμα του αυτοσυζυγούς τελεστή Τ αποτελείται από τα σημεία στα οποία η μ(χ) δεν είναι σταθερή.…”

Section: προλογος δευτερης εκδοσηςunclassified

“…Αποδεικνύεται και το αντίστροφο του Πορίσματος 2.1 ( βλέπε [ 105 ], [ 99 ], [ 2 ], [ 5 ] ) ότι, δηλαδή, εάν ο Γ είναι αυτοσυζυγής τότε το μέτρο ορθογωνιότητας είναι μοναδικό.…”

Section: Ja Jdunclassified

“…Εάν οι ακολουθίες των συντελεστών c n και λ η στη θεμελιώδη αναδρομική σχέση (6.1.1) συγκλίνουν προς πεπερασμένα όρια e και λ, αντίστοιχα, τόσο γρήγορα όσο το η- 2 , δηλαδή εάν συμβαίνει c n -c = 0(rc-2 ), λ"-λ = 0(η-2 ), ( Τότε η ελάχιστη (μέγιστη) ιδιοτιμή λ^ν) είναι κοίλη (κυρτή) συνάρτηση ως προς λ, δηλαδή λ'»<0 (λ»>0).…”

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Ορθογωνια Πολυωνυμα Και Σχετιζομενα Με Αυτα Προβληματα Της Συναρτησιακης Αναλυσεως

Παναγόπουλος¹

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Non‐classical Tauberian and Abelian type criteria for the moment problem

Patie

Vaidyanathan

2022

Mathematische Nachrichten

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The aim of this paper is to provide some new criteria for the determinacy problem of the Stieltjes moment problem. We first give a Tauberian type criterion for moment indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for moment determinacy. We also provide a criterion for moment determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an Abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to the Pedersen's refinement of the celebrated Krein's theorem. The proofs utilize non-classical Tauberian results for moment sequences that are analogues to the ones developed in [8] and [3] for the bi-lateral Laplace transforms in the context of asymptotically parabolic functions. We illustrate our results by studying the time-dependent moment problem for the law of log-Lévy processes viewed as a generalization of the log-normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally-negative Lévy processes having a Gaussian component, which may be of independent interest.

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The Classical Moment Problem and Some Related Questions in Analysis

Cited by 507 publications

References 0 publications

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Ορθογωνια Πολυωνυμα Και Σχετιζομενα Με Αυτα Προβληματα Της Συναρτησιακης Αναλυσεως

Non‐classical Tauberian and Abelian type criteria for the moment problem

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