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Selvaraj , Lacunary interpolation by cosine polynomials, Ann. Jakimovski , Reflections onWalsh equiconvergence and equisummability, Acta Univ. Lodziensis Folia Math. Lee and H.

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Tan , Spline interpolation and wavelet construction, Appl. Szabados , Birkhoff type interpolation on perturbed roots of unity, in: N. Govil et al Eds. Varma Memorial Volume, Monogr. Textbooks Pure Appl. Jakimovski , Hermite interpolation on Chebyshev nodes and Walsh equiconvergence, in: N. Jakimovski , Hermite interpolation on Chebyshev nodes and Walsh equiconvergence, J.

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Ziegler , Hermite interpolation on some perturbed roots of unity, Analysis Munich 19 1 1— Some simple properties of up x , in: Z. Ahmad, N.

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Jakimovski , Hermite interpolation on Chebyshev nodes and Walsh equiconvergence. II, J. Chui and Larry L. Singapore, , pp. Algorithms 23 1 — Algorithms 25 — Bokhari and H. Dikshit , Birkhoff interpolation on perturbed roots of unity: revisited, Numer. Algorithms 25 47— Jakimovski , Quantitative results on equiconvergence of certain sequences of rational interpolants, Math.

Cavaretta , Optimal recovery of interpolation operators in Hardy spaces, Adv. Numerous extensions of Markov inequality for various families of univariate and multivariate polynomials are known. For an overview of univariate inequalities see [2] or [3]; a survey of multivariate Markov-type inequalities can be found in [6]. For the sharpness of this upper bound see [5] and [7]. Thus, the rates of Markov Factors for homogeneous polynomials are substantially smaller than for ordinary polynomials.

It is also shown in [5] that for smooth 0-symmetric convex bodies the log n in 1. In all of the papers on homogeneous polynomials mentioned above the symmetry of the domain played an essential role. The consideration of nonsymmetric domains K will require a more delicate study of the geometry of K around the origin. Also we shall relax the assumption of convexity of the domain and replace it by the more general starlike property.

On the other hand the left inequality requires that r x does not vanish at 0 exponentially. Theorem 1. Note that condition 1. Now let us address the question of sharpness of estimate 1. Moreover, by 1. Then by 2. Lemma 2. Therefore, combining estimates 2. First, observe that condition 2. Markov-Type Inequalities 11 Thus, by 2. Using Lemma 2.

Note that property 2. K Moreover, by 2. Thus, in conditions 1.

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Without loss of generality we can assume supporting line of K that l is a coordinate axis this can be achieved by a rotation ; i. This completes the proof of the Theorem 1. References [1] R.

Borwein, T. Louis, MO, , pp. Harris, A Bernstein-Markov theorem for normed spaces, J.

Mod-05 Lec-21 Interpolation

Markov-Type Inequalities 15 [5] A. Chui, M. Neamtu, L. Schumaker, eds. Revesz, Y. Sarantopoulos, On Markov constants of homogeneous polynomials over real normed spaces, East J. The Remez inequality S and is clearly sharp, as has the same form as 1. Question 1. Videnski [38] and T.

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Erdelyi, respectively; see the references in the book [3]. The results discussed pose several problems for multivariate polynomials. Of course, it would be unlikely to establish such inequalities in a sharp form. So we are mostly looking for a sharp order of growth as the relative measure tends to zero. According to the one-dimensional results exhibited above Local Ineqalities for Multivariate Polynomials 19 we have the following groups of problems for multivariate polynomials. We begin with a Chebyshev type problem. Problem 2. The next is a Remez type problem formulated as follows. Conjecture 2.

Otherwise, the conjecture remains an open problem. The problems presented have had a lot of generalizations and variants. We mention only several of them. One-dimensional prototypes of this problem are those of Cartwright and of Plancherel-Polya. Therefore we will restrict our consideration to some results studied in the authors papers and results related to them. Based on a simple geometric fact proved in [36] it is established in [4] that the optimal constant of Problem 2. In some cases equality 3.

Formula 3. Strikingly, the optimal constant of Problem 2. Estimate 3. The very same proof gives inequality 3. To explain the relation of 3. We present here only two such results referring the interested reader to the papers [9]-[13], [15], [17]. Moreover, the proof in [9] gives some information on the size of N. Then the following is true: Theorem 3.