CHEBYSHEV POLYNOMIALS AND MARKOV-BERNSTEIN TYPE INEQUALITIES FOR RATIONAL SPACES

This paper considers the trigonometric rational system {1, 1 +/- sin t /cos t-a(1), 1 +/- sin t/cos t-a(2),...} on R(mod 2 pi) and the algebr aic rational system {1, 1/x-a(1), 1/x-a(2),...} on the interval [-1, 1 ] associated with a sequence poles (a(k))(k-1)(infinity) in R\[-1, 1]. Chebyshev polynomials for the rational trigonometric system are expli citly found. Chebyshev polynomials of the first and second kinds for t he algebraic rational system are also studied, as well as orthogonal p olynomials with respect to the weight function (1-x(2))(-1/2). Notice that in these situations, the 'polynomials' are in fact rational funct ions. Several explicit expressions for these polynomials are obtained. For the span of these rational systems, an exact Bernstein-Szego type inequality is proved, whose limiting case gives back the classical Be rnstein-Szego inequality for trigonometric and algebraic polynomials. It gives, for example, the sharp Bernstein-type inequality [GRAPHICS] where p is any real rational function of type (n, n) with poles a(k) i s an element of R\[-1, 1]. An asymptotically sharp Markov-type inequal ity is also established, which is at most a factor of 2n/(2n-1) away f rom the best possible result. With proper interpretation of root(a(k)( 2)-1), most of the results are established for (a(k))(k-1)infinity in C\[-1, 1] in a more general setting.