Chebyshev Polynomials and Markov–Bernstein Type Inequalities for Rational Spaces
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This paper considers the trigonometric rational system on the interval [1, 1] associated with a sequence of distinct poles Chebyshev polynomials for the rational trigonometric system are explicitly found. Chebyshev polynomials of the first and second kinds for the algebraic rational system are also studied, as well as orthogonal polynomials with respect to the weight function. Notice that in these situations, the “polynomials” are in fact rational functions. Several explicit expressions for these polynomials are obtained. For the span of these rational systems, an exact Bernstein-Szeg type inequality is proved, whose limiting case gives back the classical Bernstein-Szeg inequality for trigonometric and algebraic polynomials. It gives, for example, the sharp Bernstein-type inequality where p is any real rational function of type (n, n) with poles a k. An asymptotically sharp Markov-type inequality is also established, which is at most a factor of 2n/(2n 1) away from the best possible result. With proper interpretation of in a more general setting. © 1994 The London Mathematical Society.
author list (cited authors)
Borwein, P., Erdélyi, T., & Zhang, J.