Rational System Of Polynomial Interpolation

In this paper we are interested in polynomial interpolation in the rational systems. It can be divided into three chapters.In chapter one,the present situation of polynomial interpolation in the rational systems is summarized. Many extremal problems are solved by the Cheby- \ shev polynomials and the density of the Markov spaces is also closely relate the Chebyshev polynomial.Chebyshev polynomials are ubiquitous and have many applications. So the study of Chebyshev polynomial is meaningful. In this paper we discussed the polynomial interpolation based on the the zero of the Chebyshev polynomials in the rational systems. It is also meaningful.In chapter two, we discussed Lagrange interpolation based on zeros of the Chebyshev polynomial for the rational systems {1, 1/(x-a1),..., l/(x-an)},with distinct real poles{ak}nk=1 R\[-1,1]. The Lagrange interpolation is in the rational system {1, 1/(x-a1),..., 1/(x-an-1)} The corresponding Lebesgue constant is estimated. The order is shown to be asymptotically of In n when the poles stay outside an interval which contains [-1,1] in its interior. Some well-known results of classical polynomial interpolation are extended. We also gave a quadrature formula in the rational systems.In chapter three, we discussed Hermite-Fejer interpolation and Griinwald interpolation based on zeros of the Chebyshev polynomials for the real rational system Pn(a1,..., an) with the nonreal poles in {ak}nk=1 C\[-1,1]. We conclude that the corresponding Hermite-Fejer interpolation converges uniformly for every continuous function on [-1,1] provided that the poles stay outside a circle which contains the unit in its interior. We also characterize the uniform approximation of the corresponding Griinwald interpolation. This extends some well-known results of Fejer and Griinwald for the classical polynomial case.