Markov and Bernstein Inequalities in Lp for Some Weighted Algebraic and Trigonometric Polynomials

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 BernsteinErdös 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 Lp norms on the same spaces.