© 1994 by London Mathematical Society
© The London Mathematical Society
Chebyshev Polynomials and Markov–Bernstein Type Inequalities for Rational Spaces
Department of Mathematics, Simon Fraser University Burnaby, British Columbia, Canada V5A 156 E-mail: pborwein{at}cs.sfu.ca, erdelyi{at}cs.sfu.ca
Department of Computer Science, Stanford University Stanford, California 94305, USA E-mail: zhang{at}sccm.stanford.edu
Received 7 July 1992. Revision received 25 March 1993.
This paper considers the trigonometric rational system
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) and the algebraic rational system
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in R\[–1, 1]. 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 (1 – x2)–1/2. 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-Szegö type inequality is proved, whose limiting case gives back the classical Bernstein-Szegö inequality for trigonometric and algebraic polynomials. It gives, for example, the sharp Bernstein-type inequality
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R\[–1, 1]. 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 
, most of the results are established for in C\[–1, 1] in a more general setting.
This material is based upon work supported by NSERC of Canada (P.B.) and National Science Foundation under Grant No. DMS-9024901 (T.E.).