On Bernstein and Markov-Type Inequalities for Multivariate Polynomials on Convex Bodies

Kroó, András; Révész, Szilárd Gy.

doi:10.1006/jath.1998.3314

Cited by 44 publications

(36 citation statements)

References 7 publications

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“…[26]. Consider a cartesian grid {(ih, jh) , i, j ∈ Z} with constant stepsize h: for every square of the grid that has nonempty intersection with K, take a point in this intersection.…”

Section: A Circular Sectormentioning

confidence: 99%

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Bos¹,

Marchi²,

Sommariva³

et al. 2010

SIAM J. Numer. Anal.

117

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We discuss and compare two greedy algorithms, that compute discrete versions of Fekete-like points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the so-called "Approximate Fekete Points" by QR factorization with column pivoting of Vandermonde-like matrices. The second computes Discrete Leja Points by LU factorization with row pivoting. Moreover, we study the asymptotic distribution of such points when they are extracted from Weakly Admissible Meshes.2000 AMS subject classification: 41A10, 41A63, 65D05.

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“…[26]. Consider a cartesian grid {(ih, jh) , i, j ∈ Z} with constant stepsize h: for every square of the grid that has nonempty intersection with K, take a point in this intersection.…”

Section: A Circular Sectormentioning

confidence: 99%

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Bos¹,

Marchi²,

Sommariva³

et al. 2010

SIAM J. Numer. Anal.

117

View full text Add to dashboard Cite

show abstract

“…Note that in [4] the best ellipse is not found; the construction there gives only a good estimate, but not an exact value of (2.16) or (2.17). In fact, here we quoted [4] in a strengthened form: the original paper contains a somewhat weaker formulation only.…”

Section: B Milev and Sz Gy Révész 149mentioning

confidence: 93%

“…Theorem 2.7 (Kroó-Révész [4]). Let K be an arbitrary convex body, x ∈ intK and y = 1, where X can be an arbitrary normed space.…”

Section: B Milev and Sz Gy Révész 149mentioning

confidence: 99%

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Bernstein's inequality for multivariate polynomials on the standard simplex

Milev

Révész

2005

J Inequal Appl

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The classical Bernstein pointwise estimate of the (first) derivative of a univariate algebraic polynomial on an interval has natural extensions to the multivariate setting. However, in several variables the domain of boundedness, even if convex, has a considerable geometric variety. In 1990, Y. Sarantopoulos satisfactorily settled the case of a centrally symmetric convex body by a method we may call "the method of inscribed ellipses." On the other hand, for the general case of nonsymmetric convex bodies we are only within a constant factor of an exact inequality. The best known results suggest relevance of the generalized Minkowski functional, and a natural conjecture for the exact Bernstein factor was formulated with this geometric quantity. This work deals with the most natural and simple nonsymmetric case, that of a standard simplex in R d , and computes the exact yield of the method of inscribed ellipses. Although the known general estimates of the Bernstein factor are improved for the simplex here, we find that not even the exact yield of the inscribed ellipse method reaches the conjecture. However, we also show that for an arbitrary convex body the subset of ridge polynomials satisfies the conjecture.

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“…There is now such extensive literature on Markov type inequalities that it is beyond the scope of this paper to give a complete bibliography. Let us mention only certain works which are most closely related to our paper (with emphasis on those dealing with generalizations of Markov's inequality on sets admitting cusps), for example [1][2][3][4][5][6][7]35,36,39,43,44,46,48,50,51,57,58]. We should stress here that the present paper owes a great debt particularly to Pawłucki and Pleśniak's work, because in [43] they laid the foundations for the theory of polynomial inequalities on "tame" (for example, semialgebraic) sets with cusps.…”

Section: Theorem 11 (Markov) If P Is a Polynomial Of One Variable Thenmentioning

confidence: 99%

Markov’s inequality and polynomial mappings

Pierzchała

2015

Math. Ann.

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Markov's inequality is a certain estimate for the norm of the derivative of a polynomial in terms of the degree and the norm of this polynomial. It has many interesting applications in approximation theory, constructive function theory and in analysis (for instance, to Sobolev inequalities or Whitney-type extension problems). One of the purposes of this paper is to give a solution to an old problem, studied among others by Baran and Pleśniak, and concerning the invariance of Markov's inequality under polynomial mappings (polynomial images). We also address the issue of preserving Markov's inequality when taking polynomial preimages. Lastly, we give a sufficient condition for a subset of a Markov set to be a Markov set.

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On Bernstein and Markov-Type Inequalities for Multivariate Polynomials on Convex Bodies

Cited by 44 publications

References 7 publications

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Bernstein's inequality for multivariate polynomials on the standard simplex

Markov’s inequality and polynomial mappings

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