Polynomial inequalities on general subsets of $R^N$

Pierre Goetgheluck

Displaying similar documents to “Polynomial inequalities on general subsets of $R^{N}$ ”

Sets with the Bernstein and generalized Markov properties

Mirosław Baran, Agnieszka Kowalska (2014)

Annales Polonici Mathematici

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It is known that for $C^{\infty}$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty}$ determining. In this paper we give examples of sets which are not $C^{\infty}$ determining, but have the Bernstein and generalized Markov properties.

Tangential Markov inequality in $L^{p}$ norms

Agnieszka Kowalska (2015)

Banach Center Publications

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In 1889 A. Markov proved that for every polynomial p in one variable the inequality $| | p^{'} {| |}_{[- 1, 1]} {\leq (d e g p) ² | | p | |}_{[- 1, 1]}$ is true. Moreover, the exponent 2 in this inequality is the best possible one. A tangential Markov inequality is a generalization of the Markov inequality to tangential derivatives of certain sets in higher-dimensional Euclidean spaces. We give some motivational examples of sets that admit the tangential Markov inequality with the sharp exponent. The main theorems show that the results on certain arcs...

Markov's property of the Cantor ternary set

Leokadia Białas, Alexander Volberg (1993)

Studia Mathematica

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We prove that the Cantor ternary set E satisfies the classical Markov inequality (see [Ma]): for each polynomial p of degree at most n (n = 0, 1, 2,...) (M) $| p^{'} (x) | \leq M n^{m} s u p_{E} | p |$ for x ∈ E, where M and m are positive constants depending only on E.

Markov's property for kth derivative

Mirosław Baran, Beata Milówka, Paweł Ozorka (2012)

Annales Polonici Mathematici

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Consider the normed space $(ℙ (ℂ^{N}), | | \cdot | |)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_{g} : (ℙ (ℂ^{N}), | | \cdot | |) ∋ f \mapsto g f \in (ℙ (ℂ^{N}), | | \cdot | |)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $| \partial / \partial z_{j} {P | | \leq M (d e g P)}^{m} | | P | |$ , j = 1,...,N, $P \in ℙ (ℂ^{N})$ , with positive constants M and m is equivalent to the inequality $| | \partial^{N} / \partial z ₁ . . . \partial z_{N} P | | \leq M^{'} {(d e g P)}^{m^{'}} | | P | |$ , $P \in ℙ (ℂ^{N})$ , with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras....

Compact subsets of $C^{n}$ preserving Markov’s inequality

W. Plesniak (1988)

Matematički Vesnik

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Eigenvalues of operators in $L_{p}$ -spaces in Markov chains with a general state space

Zbyněk Šidák (1967)

Czechoslovak Mathematical Journal

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The Nagaev-Guivarc’h method via the Keller-Liverani theorem

Loïc Hervé, Françoise Pène (2010)

Bulletin de la Société Mathématique de France

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The Nagaev-Guivarc’h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish limit theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. The paper outlines this method and extends it by stating a multidimensional local limit theorem, a one-dimensional Berry-Esseen theorem, a first-order...

Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$

Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E \subset ℂ^{N}$ . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_{E}$ . Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $l i m_{n \to \infty} M ₙ {(E)}^{1 / n} = 1$ where $M ₙ (E) : = s u p {| | | g r a d P | | |}_{E} / {| | P | |}_{E}$ , the supremum is taken over all polynomials P of N variables...

Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^{d}$

Piotr Bugiel (1998)

Annales Polonici Mathematici

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Asymptotic properties of the sequences (a) ${P_{φ}^{j} g}_{j = 1}^{\infty}$ and (b) ${j^{- 1} \sum_{i = 0}^{j - 1} P ⁱ_{φ} g}_{j = 1}^{\infty}$ , where $P_{φ} : L ¹ \to L ¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov...

The scaling limits of a heavy tailed Markov renewal process

Julien Sohier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the $α$ -stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0, \infty) \times [0, a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality.

Two stationarity conditions in the $G I / r_{0} / M / r_{1}$ dam

M. Jankiewicz (1988)

Applicationes Mathematicae

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Mean lower bounds for Markov operators

Eduard Emel&#039;yanov, Manfred Wolff (2004)

Annales Polonici Mathematici

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Let T be a Markov operator on an L¹-space. We study conditions under which T is mean ergodic and satisfies dim Fix(T) < ∞. Among other things we prove that the sequence $(n^{- 1} \sum_{k = 0}^{n - 1} T^{k}) ₙ$ converges strongly to a rank-one projection if and only if there exists a function 0 ≠ h ∈ L¹₊ which satisfies $l i m_{n \to \infty} | | (h - n^{- 1} \sum_{k = 0}^{n - 1} T^{k} f) ₊ | | = 0$ for every density f. Analogous results for strongly continuous semigroups are given.

An optimal strong equilibrium solution for cooperative multi-leader-follower Stackelberg Markov chains games

Kristal K. Trejo, Julio B. Clempner, Alexander S. Poznyak (2016)

Kybernetika

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This paper presents a novel approach for computing the strong Stackelberg/Nash equilibrium for Markov chains games. For solving the cooperative $n$ -leaders and $m$ -followers Markov game we consider the minimization of the $L_{p} -$ norm that reduces the distance to the utopian point in the Euclidian space. Then, we reduce the optimization problem to find a Pareto optimal solution. We employ a bi-level programming method implemented by the extraproximal optimization approach for computing the strong...

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou (2017)

Czechoslovak Mathematical Journal

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Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $∥ p^{'} ∥_{[- 1, 1]} \leq \frac{1}{2} {∥ p ∥}_{[- 1, 1]}$ for a constrained polynomial $p$ of degree at most $n$ , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(- 1, 1)$ and establish a new asymptotically sharp inequality. ...

An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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The main result says in particular that if $t (ζ) : = \sum_{ν = - n} ⁿ c_{ν} e^{i ν ζ}$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.

On realizability of sign patterns by real polynomials

Vladimir Kostov (2018)

Czechoslovak Mathematical Journal

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p, n)$ , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials. ...

Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

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Let E be a compact set in the complex plane, $g_{E}$ be the Green function of the unbounded component of $ℂ_{\infty} ∖ E$ with pole at infinity and $M ₙ (E) = s u p (| | P^{'} {| |}_{E} {) / (| | P | |}_{E})$ where the supremum is taken over all polynomials ${P |}_{E} ≢ 0$ of degree at most n, and ${| | f | |}_{E} = s u p | f (z) | : z \in E$ . The paper deals with recent results concerning a connection between the smoothness of $g_{E}$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${M ₙ (E)}_{n = 1, 2, . . .}$ . Some additional conditions are given for special classes of sets.

Evaluating default priors with a generalization of Eaton’s Markov chain

Brian P. Shea, Galin L. Jones (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $𝛷$ be a class of functions on the parameter space and consider estimating elements of $𝛷$ under quadratic loss. If the formal Bayes estimator of every function in $𝛷$ is admissible, then the prior is strongly admissible with respect to $𝛷$ . Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with...

Approximation by weighted polynomials in $ℝ^{k}$

Maritza M. Branker (2005)

Annales Polonici Mathematici

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We apply pluripotential theory to establish results in $ℝ^{k}$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $Σ \subset ℝ^{k}$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

Displaying similar documents to “Polynomial inequalities on general subsets of R N ”

Displaying similar documents to “Polynomial inequalities on general subsets of $R^{N}$ ”