Markov–Bernstein Type Inequalities for Multivariate Polynomials on Sets with Cusps

Kroó, András; Szabados, J.

doi:10.1006/jath.1999.3378

Cited by 19 publications

(22 citation statements)

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“…This Markov type estimate can be found in various papers (see e.g. [11] for corresponding references). The fact that a Markov type inequality of order O(n 2 α ) is sharp in general for Lipα domains goes back to an earlier paper by Goetgheluck [7].…”

Section: Remark 1 Bernstein Type Inequalitymentioning

confidence: 92%

Bernstein type inequalities on star-like domains in $${\mathbb{R }}^d$$ with application to norming sets

Kroó¹

2013

Bull. Math. Sci.

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n be the space of real algebraic polynomials of d variables and degree at most n, K ⊂ R d a compact set,In this note we shall prove that if K is a star-like domain with Lipα boundary, that is ϕ K (x) satisfies the Lipα condition, 0 < α ≤ 1 then the following Bernstein type inequality holds:where |∇ p| stands for the Euclidean length of the gradient of p. Furthermore, if 1 < α ≤ 2 and K is a C α star like-domain, that is ∇ϕ K (x) has the Lip(α − 1) property, then the same inequality holds for the tangential derivatives of p. and card(Y n ) ≤ c 2 n d , n ∈ N. It was proved earlier that optimal admissible meshes exist in C 2 star-like domains. In this paper it will be shown that C 2− 2 d smoothness also suffices for their existence.

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Section: Remark 1 Bernstein Type Inequalitymentioning

confidence: 92%

Bernstein type inequalities on star-like domains in $${\mathbb{R }}^d$$ with application to norming sets

Kroó¹

2013

Bull. Math. Sci.

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“…We note that there is still active research regarding the constant in the inequality (see [12] and references therein).…”

Section: )mentioning

confidence: 99%

On the equivalence of the modulus of smoothness and the K-functional over convex domains

Dekel¹

2010

Journal of Approximation Theory

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It is well known that for any bounded Lipschitz graph domain Ω ⊂ R d , r ≥ 1 and 1 ≤ p ≤ ∞ there exist constants C 1 (d, r ) , C 2 (Ω , d, r, p) > 0 such that for any function f ∈ L p (Ω ) and t > 0where ω r ( f, ·) p is the modulus of smoothness and K r ( f, ·) p is the K -functional, both of order r . As can be seen, the right hand side inequality depends on the geometry of the domain. One of our main results is that there exists an absolute constant C 3 (d, r, p) such that for any convex domain Ω ⊂ R d and all functions f ∈ L p (Ω ), 1 ≤ p ≤ ∞,For bounded convex domains, the above estimate can be improved for 'large' values of t K r f, t r p ≤ C 4 (d, r, p) 1 − t r diam (Ω ) r µ (Ω , t) −(r −1+1/ p) + 1 ω r ( f, t) p , 0 < t ≤ diam (Ω ) .

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“…It is known that the L ∞ -result M n,∞ (K ) = O(n 2/α ) is in general asymptotically sharp for C α -domains, see [9]. This means that the L q -Markov factors of these domains also cannot be of smaller order than n 2/α .…”

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confidence: 96%

“…The Markov Factors of cuspidal domains are relatively well studied in case of uniform norm, see e.g. [11,9]. In particular if K is a star-like C α -domain then it is known that M n,∞ (K ) = O(n 2/α ).…”

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confidence: 99%

On Bernstein–Markov-type inequalities for multivariate polynomials in Lq-norm

Kroó

2009

Journal of Approximation Theory

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Bernstein-Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein-Markov-type inequalities in the L q -norm, and it requires introduction of new methods. In this paper we verify certain Bernstein-Markov-type inequalities in L q -norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.Let us introduce some basic notations used in this paper. We shall denote by S d−1 the unit sphere in R d , B d (a, r ) := {x ∈ R d : |a − x| ≤ r } stands for the ball centered at a ∈ R d and radius r . For a convex body K ∈ R d denote by r K the so-called width of K , which is defined as the radius of the largest ball contained in K . In case if K is a convex body it is known that it contains a unique ellipsoid E K of maximal volume, which is called the maximal ellipsoid of K .

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Markov–Bernstein Type Inequalities for Multivariate Polynomials on Sets with Cusps

Abstract: In this paper we give sharp Markov Bernstein type inequalities for derivatives of multivariate polynomials for a wide family of domains with cusps. AcademicPress

Cited by 19 publications

References 5 publications

Bernstein type inequalities on star-like domains in $${\mathbb{R }}^d$$ with application to norming sets

Bernstein type inequalities on star-like domains in $${\mathbb{R }}^d$$ with application to norming sets

On the equivalence of the modulus of smoothness and the K-functional over convex domains

On Bernstein–Markov-type inequalities for multivariate polynomials in Lq-norm

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