Bernstein-type inequalities for the derivatives of rational functions in $L\_\{p\}$-spaces, $0&lt;p&lt;1$, on Lavrent{’}ev curves

Пекарский, Александр Антонович

doi:10.1090/s1061-0022-05-00864-2

St. Petersburg Math. J.

2005

DOI: 10.1090/s1061-0022-05-00864-2

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Bernstein-type inequalities for the derivatives of rational functions in $L\_\{p\}$-spaces, $0<p<1$, on Lavrent{’}ev curves

Александр Антонович Пекарский

Abstract: Let S be a simple or a closed Lavrent ev curve on the complex plane, let 0 < p < 1 with 1/p ∈ N, and let s ∈ N. It is shown that for an arbitrary rational function r of degree n such that |r| p is integrable on S the following inequality is fulfilled:

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Cited by 2 publications

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Estimates for derivatives of rational functions and the fourth Zolotarev problem

Lukashov

2008

St. Petersburg Math. J.

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Abstract. An estimate is obtained for the derivatives of real rational functions that map a compact set on the real line to another set of the same kind. Many well-known inequalities (due to Bernstein, Bernstein-Szegő, V. S. Videnskiȋ, V. N. Rusak, and M. Baran-V. Totik) are particular cases of this estimate. It is shown that the estimate is sharp. With the help of the solution of the fourth Zolotarev problem, a class of examples is constructed in which the estimates obtained turn into identities. §1. IntroductionThe Bernstein-Markov inequalitites for derivatives of trigonometric and algebraic polynomials, as well as their various generalizations, play an important role in approximation theory (see, e.g., the books [1, 2, 3] and also the recent papers [4,5,6,7] and the references therein).Our goal in the present paper is to obtain an inequality estimating the derivative of a real rational function that maps a given compact set E ⊂ R to another compact set F ⊂ R. We show that this inequality is sharp for any F and that it turns into the identity for a wide class of rational functions in the case where F is the union of two segments.We consider rational functions of degree n of the formwhere P n and Q n are real polynomials of degree at most n. Let x 1 , . . . , x n be the zeros of Q n (if deg Q n = n − k < n, we assume that x n−k+1 = · · · = x n = ∞). We shall use the harmonic measure ω(z, e, Ω) of a set e ⊂ E = ∂Ω at a point z relative to a domain Ω = C\E, and also the corresponding density). Next, we denote by Z n (x) the solution of the fourth Zolotarev problem (see [8,9,10]) concerning the best approximation of the function sgn x on E = [−1, κ]∪[κ, 1], 0 < κ < 1, by real rational functions of degree at most n.We cite one of the main results of [7].

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Estimates for derivatives of rational functions and the fourth Zolotarev problem

Lukashov

2008

St. Petersburg Math. J.

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Sublinear dimension growth in the Kreiss Matrix Theorem

Nikolski

2014

St. Petersburg Math. J.

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A possible sublinear dimension growth in the Kreiss Matrix Theorem, bounding the stability constant in terms of the Kreiss resolvent characteristic, is discussed. Such a growth is proved for matrices having unimodular spectrum and acting on a uniformly convex Banach space. The principal ingredients to results obtained come from geometric properties of eigenvectors, where the approaches by C. A. McCarthy-J. Schwartz (1965) and V. I. Gurarii-N. I. Gurarii (1971) are used and compared. The sharpness issue is verified via finite Muckenhoupt bases (by using mostly the approach by M. Spijker, S. Tracogna, and B. Welfert (2003)).

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Bernstein-type inequalities for the derivatives of rational functions in $L\_\{p\}$-spaces, $0<p<1$, on Lavrent{’}ev curves

Abstract: Let S be a simple or a closed Lavrent ev curve on the complex plane, let 0 < p < 1 with 1/p ∈ N, and let s ∈ N. It is shown that for an arbitrary rational function r of degree n such that |r| p is integrable on S the following inequality is fulfilled:

Cited by 2 publications

References 11 publications

Estimates for derivatives of rational functions and the fourth Zolotarev problem

Estimates for derivatives of rational functions and the fourth Zolotarev problem

Sublinear dimension growth in the Kreiss Matrix Theorem

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