Remez-type inequality on sets with cusps

Pierzchała, Rafał

doi:10.1016/j.aim.2015.03.028

Cited by 9 publications

(3 citation statements)

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“…Inequalities of the form (5.5) are known also for sets Z of measure zero, for discrete or finite Z (see [16,82,83] and references therein). Similar inequalities have been studied for restrictions of polynomials to semi-algebraic (subanalytic) sets ([2]- [4], [11,11,13,15,23,24,53,83]). However, in contrast with Theorem 5.7, already on algebraic curves we cannot hope to get a uniform bound, depending only on the degree and on the geometry (measure) of Z.…”

Section: )mentioning

confidence: 93%

“…In this section we briefly discuss robustness of polynomial approximation on algebraic curves. This is an important question in Approximation Theory, and its connection with a sort of analytic parametrization is well known (see [2]- [4], [53,54,55,83] and references therein). One of the main tools here is provided by "Remez-type" or "norming" inequalities, which compare maximum of a polynomial on the unit interval I = [−1, 1] with its maximum on a given subset Z ⊂ I.…”

Section: Remez-type Inequalities On Algebraic Curvesmentioning

confidence: 99%

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Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

Yomdin

2015

Japan J. Indust. Appl. Math.

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Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently rather separated domains: smooth dynamics, diophantine geometry, approximation theory, and computational geometry. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the paper. We present also some new results, stressing interconnection between various types and various applications of smooth parametrization.

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Section: )mentioning

confidence: 93%

Section: Remez-type Inequalities On Algebraic Curvesmentioning

confidence: 99%

Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

Yomdin

2015

Japan J. Indust. Appl. Math.

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“…Markov type inequalities and related topics have been studied by many authors; see for instance [1][2][3][6][7][8][9][10][11][12][13][14][15][19][20][21]24,29,30,32,34,35,38,40,47]. In this paper, we are interested in the following problem.…”

Section: Has the Hcp Property Andk ⊂ U Under What Conditions Does Imentioning

confidence: 99%

Geometry of holomorphic mappings and Hölder continuity of the pluricomplex Green function

Pierzchała

2020

Math. Ann.

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We provide a solution to a long-standing open problem that lives in the interface of pluripotential theory and multivariate approximation theory. The problem is to characterize the holomorphic maps which preserve Hölder continuity of the pluricomplex Green function associated with a compact subset of C N . We also prove, under mild restrictions, that nondegenerate holomorphic maps preserve Markov's inequality for polynomials.Geometry of holomorphic mappings... Definition 1.2 (see [29]) We say that a compact set ∅ = K ⊂ C N has the HCP property if there exist , μ > 0 such that, for each z ∈ K (1) ,(1.4)In the above definition, and subsequently, we use the following notation: for each set ∅ = A ⊂ C N and each r > 0, we put A (r ) := {z ∈ C N : dist(z, A) ≤ r } .

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The Sharp Remez-Type Inequality for Even Trigonometric Polynomials on the Period

Erdélyi

2019

Topics in Classical and Modern Analysis

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We prove that max t∈ [−π,π] |Q(t)| ≤ T 2n (sec(s/4)) = 1 2 ((sec(s/4) + tan(s/4)) 2n + (sec(s/4) − tan(s/4)) 2n ) for every even trigonometric polynomial Q of degree at most n with complex coefficients satisfyingwhere m(A) denotes the Lebesgue measure of a measurable set A ⊂ R and T 2n is the Chebysev polynomial of degree 2n on [−1, 1] defined by T 2n (cos t) = cos(2nt) for t ∈ R. This inequality is sharp. We also prove that max t∈[−π,π] |Q(t)| ≤ T 2n (sec(s/2)) = 1 2 ((sec(s/2) + tan(s/2)) 2n + (sec(s/2) − tan(s/2)) 2n )for every trigonometric polynomial Q of degree at most n with complex coefficients satisfying ForewordI started my Ph.D. program in June, 1987, at The Ohio State University, as a student of Paul Nevai. In August, 1987, Paul Nevai moved from Columbus, Ohio, to Columbia, South Carolina, to spend the school year 1987-88 at the University of South Carolina, and each of his students followed him. This is how I met Mr. (Yingkang) Hu, who was a student of Ron DeVore at that time. It was a somewhat short but very exciting time for me at the University of South Carolina. Not only had I passed each of my necessary exams in my Ph.D. program but I had the possibility to learn from and interact with some of the very best researchers in approximation theory. My office neighbors were Paul Nevai, Ron DeVore, and George Lorenz. George Lorentz happened to be Ron DeVore's Key words and phrases. trigonometric polynomials, Remez-type inequalities, geometry of polynomials.

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Remez-type inequality on sets with cusps

Abstract: A Remez-type inequality is proved for a large family of sets with cusps in R N , including compact, fat and semialgebraic (or subanalytic) sets.

Cited by 9 publications

References 58 publications

Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

Geometry of holomorphic mappings and Hölder continuity of the pluricomplex Green function

The Sharp Remez-Type Inequality for Even Trigonometric Polynomials on the Period

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