Extending positive definiteness

Cichoń, Dariusz; Stochel, Jan; Szafraniec, Franciszek Hugon

doi:10.1090/s0002-9947-2010-05268-7

Cited by 19 publications

(23 citation statements)

References 52 publications

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“…The issue we deal with below can be regarded as the truncated complex moment problem; however, it deviates from the usual meaning of that term in which only finite systems are considered. The interested reader can consult [5] for the discussion relating Theorem 1 to a recent result on the truncated complex moment problem contained in [6]. .…”

Section: Prefatory Mattersmentioning

confidence: 99%

“…Other matters regarding Theorem 1 (including the case of non-symmetric T 's) are also exhibited in [5].…”

Section: Prefatory Mattersmentioning

confidence: 99%

“…In the recent paper [5] we have proved a version of the Riesz-Haviland theorem solving the truncated moment problem in which polynomials are built up from the selected monomials z mzn with (m, n) ranging over an index set T ; however, the positivity condition has to be assured on the whole complex plane (cf. Theorem 1).…”

Section: Introductionmentioning

confidence: 99%

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Riesz–Haviland criterion for incomplete data

Cichoń

Stochel

Szafraniec

2011

Journal of Mathematical Analysis and Applications

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The aim of this paper is to analyze a "support-free" version of the Riesz-Haviland theorem proved recently by the present authors, which characterizes truncations of the complex moment problem via positivity condition on appropriate families of polynomials in z andz. The attention is focused on modifications of the positivity condition as well as the assumption on admissible truncations. The former results in truncations for which the corresponding "support-free" Riesz-Haviland condition locates a representing measure on the distinguished subset of the complex plane, while the latter effects a non-integral variant of the Riesz-Haviland theorem.

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Section: Prefatory Mattersmentioning

confidence: 99%

“…Other matters regarding Theorem 1 (including the case of non-symmetric T 's) are also exhibited in [5].…”

Section: Prefatory Mattersmentioning

confidence: 99%

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Riesz–Haviland criterion for incomplete data

Cichoń

Stochel

Szafraniec

2011

Journal of Mathematical Analysis and Applications

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“…There are two ways of approaching the complex moment problem (see [3]; for a recent survey of the complex moment problem see also [20]). One following an idea due to Marcel Riesz (for continuation see [12,13,14]) and the other via positive definite extendibility (see [27,9]). As is well-known, positive definiteness is not sufficient for solving the complex moment problem (see [19,3]).…”

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confidence: 99%

The complex moment problem: determinacy and extendibility

Cichoń¹,

Stochel²,

Szafraniec³

2019

Math. Scand.

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Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at 0. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through 0 is at most one point set. Further study concerns representing measures whose supports are Zariski dense in C as well as complex moment sequences which are constant on a family of parallel "Diophantine lines". All this is supported by a bunch of illustrative examples.

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“…Proposition 2 of[11] (see also [2, Proposition 6.1.8]) says that if T is a semigroup and C ⊆ T * separates points, then the functionst| C , t ∈ T are linearly independent in C C . We use this result for T = S * and C = ŝ : s ∈ S , the functionŝ σ| C can be identified with characters on S 3.…”

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confidence: 99%

Pontryagin space structure in reproducing kernel Hilbert spaces over *-semigroups

Szafraniec

Wojtylak

2010

Positivity

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The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of * -semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries.

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Extending positive definiteness

Cited by 19 publications

References 52 publications

Riesz–Haviland criterion for incomplete data

Riesz–Haviland criterion for incomplete data

The complex moment problem: determinacy and extendibility

Pontryagin space structure in reproducing kernel Hilbert spaces over *-semigroups

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