Markov and Bernstein inequalities in for some weighted algebraic and trigonometric polynomials
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Bernstein inequalities are given for polynomials of degree at most 2m (where m ≤ n), weighted by (1 + x2 )−n , in L p norms on (−∞, ∞), and also in related spaces of weighted trigonometric polynomials. Also, Bernstein and Markov inequalities valid on [0, ∞) are derived for polynomials of degree m weighted by (1 + t)−n . 1. Introduction Let Qm,n (with m ≤ n) denote the space of polynomials of degree 2m or less on (−∞, ∞), weighted by (1 + x2 )−n . The elements Qm,n are thus rational functions with denominator (1 + x2 )m and numerator of degree at most 2m (if m = n, we can write, more briefly, Qn for Qn,n ). The spaces Qm,n form a nested sequence as n increases and r = n − m is held to some given value of weighted polynomial spaces, with the weight depending upon n. As these spaces can obviously be used for approximation on the real line, their approximation-theoretic properties are worthy of a systematic investigation, of which this article is a part. Briefly describing the previous work, the properties of Lagrange interpolation in these spaces were investigated in Kilgore [2]. The main result there was that the BernsteinErd¨os conditions characterize interpolation of minimal norm into these spaces, as was already known for spaces of ordinarly polynomials, for trigonometric polynomials, and for several other classes of polynomial spaces with weighted norm. Inside of any of these rational function spaces Qm,n , there is the subspace of even functions. That subspace is isometrically isomorphic to a space defined upon the half-line [0, ∞), which has been denoted by Rm,n in Kilgore [3]. Specifically, a typical function in Rm,n is a rational function with denominator (1 + t)n and numerator Pm (t), where Pm is a polynomial of degree at most m. The natural isometry between Rm,n and the even part of Qm,n is induced by t ↔ x2 . This space had not been discussed in Kilgore [2], since the corresponding results about interpolation in the spaces Rm,n follow as a special case from Kilgore [1]. In the article [3], analogues of the Markov and Bernstein inequalities were shown to hold, both in the spaces Qm,n and in the spaces Rm,n , under the uniform norm. Here, we show Markov and Bernstein inequalities in L p norms on the same spaces. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 413–421 DOI: 10.1155/JIA.2005.413
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Weighted polynomial Bernstein inequalities in L p
It was also noted in Kilgore [2] that Qn is isometrically isomorphic to Tn , the space of trigonometric polynomials of degree at most n, via the mapping x ↔ tanθ/2. By the same mapping, there is an isometric isomorphism between the spaces Qm,n and the weighted spaces of trigonometric polynomials Tm,n , where Tm,n consists of the space Tm with weight cos2r θ/2, where r = n − m as previously mentioned. Thus, in [2] similar results for interpolation were shown for these weighted trigonometric polynomial spaces, too, and in [3] a weighted Bernstein inequality with uniform norm was shown to hold. Using the same in
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