An explicit version of the Chebyshev-Markov-Stieltjes inequalities and its applications

Hürlimann, Werner

doi:10.1186/s13660-015-0709-1

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…By the same formula, the orthonormal polynomials for , D are well-defined for degrees ≤ κ, since the orthonormal polynomials up to degree κ attached to X exist. By [13,Th. 4.1] or [21,Th.…”

Section: Distribution Functionsmentioning

confidence: 96%

Designs in finite metric spaces: a probabilistic approach

Shi

Rioul

Solé

2021

Preprint

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A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in Q-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given.

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Section: Distribution Functionsmentioning

confidence: 96%

Designs in finite metric spaces: a probabilistic approach

Shi

Rioul

Solé

2021

Preprint

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“…Furthermore, bounds on distribution functions have some importance in probability theory and statistics; and various bounds are referenced to Chebyshev, Markov, Stieltjes and others. This includes variants of the CMS Theorem formulated in terms of moments, e.g., [Zel54] or more recently [Hür15]. Moment-matching methods also appear in the context of system theory [Ant05].…”

Section: Applicationsmentioning

confidence: 99%

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

Jawecki¹

2022

Preprint

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The accumulated quadrature weights of Gaussian quadrature formulae constitute bounds on the integral over the intervals between the quadrature nodes. Classical results in this concern date back to works of Chebyshev, Markov and Stieltjes and are referred to as Separation Theorem of Chebyshev-Markov-Stieltjes (CMS Theorem). Similar separation theorems hold true for some classes of rational Gaussian quadrature. The Krylov subspace for a given matrix and initial vector is closely related to orthogonal polynomials associated with the spectral distribution of the initial vector in the eigenbasis of the given matrix, and Gaussian quadrature for the Riemann-Stieltjes integral associated with this spectral distribution. Similar relations hold true for rational Krylov subspaces. In the present work, separation theorems are reviewed in the context of Krylov subspaces including rational Krylov subspaces with a single complex pole of higher multiplicity and some extended Krylov subspaces. For rational Gaussian quadrature related to some classes of rational Krylov subspaces with a single pole, the underlying separation theorems are newly introduced here.

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“…That is, can one add additional (algebraic) conditions to items ii and iii in Proposition 3.2 to make these equivalent to the statements in Proposition 3.3? Applying the results of [5] would give such an algebraic condition, essentially saying that a sequence (a k ) ∞ k=0 satisfies any of the conditions in Proposition 3.3 if and only if the zeroes of a certain polynomial determined by (a k ) ∞ k=1 satisfies a inequality similar to that in Theorem 1.1. This however is not particularly illuminating and we refrain from pursuing this matter further here.…”

Section: Notice Thatmentioning

confidence: 99%

Tournament limits: Degree distributions, score functions and self-converseness

Thörnblad¹

2016

Preprint

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Motivated by known results for finite tournaments, we define and study the score functions of tournament kernels and the degree distributions of tournament limits. Our main theorem completely characterises those distributions that appear as the degree distribution of some tournament limit and those functions that appear as the score function of some tournament kernel. We also show that only the uniform distribution can be realised as the outdegree distribution of a unique tournament limit. Finally we define self-converse tournament limits and kernels and characterise their degree distributions and score functions.

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An explicit version of the Chebyshev-Markov-Stieltjes inequalities and its applications

Cited by 4 publications

References 17 publications

Designs in finite metric spaces: a probabilistic approach

Designs in finite metric spaces: a probabilistic approach

A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

Tournament limits: Degree distributions, score functions and self-converseness

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