Hermite-fej¨¦r Interpolation Of Several Problems

Approximation theory of functions is an important branch in modern mathematics. Theorems proved by Weierstrass in 1885 which says that the continuous function can be approximated uniformly by polynomials. With the rapid development of computer, approximation theory plays an important role both in research and application .Interpolation, as an important method of approximation, is an useful tool for processing of surveying data and tabulation of functions, and is a foundation to deduce many other numerical methods, For example, numerical integration, numerical solution of difference equations and solving of nonlinear equations etc. Following Lagrange interpolation ,the interpolation of Hermite, Hermite-Fejer were also investigated. Lagrange interpolation, Hermite interpolation , Hermite-Fejer interpolation . An important aspect of interpolation is that: for a given interpolation, find out the better infinite triangular intepolatory matrix.In this thesis, let Qm,n(f,T*,x) denote the modified (0,1, ...,m) Hermite-Fejer interpolation polynomial of f(x) ∈ C[-1,1] based on the Chebyshev nodes and {-1, 1}.That is, Qm,n(f,T*,x) is the polynomial of least degree which interpolates f(x) and has its first m derivatives vanish at each of the zeros of the mth chebyshev polynomial of the first kind and the points of { -1,1}.In chapter 2, we show that for a broad class of interpolatory matrix on [-1,1], let f(z) be nonconstant entire function,and be increasing, the sequence of polynomas induced by (0,1) Hermite-Fejer to f(z) diverges everywhere in the complex plane outside the interval of interpolation [-1,1], this result is in striking contrast to the behavior of the Lagrange interpolation polynomials.In Chapter 3 ,we generalize results of H - P.Blatt and M.Gotz, we show that...