Convergence Properties of Lagrange Interpolation Polynomials of | x | ~ ? - type Functions

This paper is made up of four chapters, which discuss mainly about the divergence of the Lagrange interpolation based on equidistant nodes and the best approximation order to function|x|α by interpolation polynomialsChapter one:this chapter discusses about the sequence of Lagrange interpolation polynomials corresponding to the function f(x)=|x|α (0<α<2) on equidistant nodes in [-1,1]. As we know, the larger the a is, the more smoothness the|x|α has. We will construct a new interpolated process for|x|α (0<α<2) to show that the sequence divergence everywhere in the interval except at zero and the end-points.Chapter two:this chapter discusses about the sequence of Lagrange interpolation polynomials corresponding to the function (0<α≤1) on equidistant nodes in [-1,1]. We will show that the sequence diverges everywhere in the interval except at zero and the end-points in the odd degree case.Chapter three:this chapter discusses about the exact rate of convergence for Lagrange interpolation polynomials at the point zero. We partialy solved M.Revers’ conjecture to f(x)=|x|α takes place for all α>1(except a is an even integer). We will give the exact convergence rate at the zero of Lagrange interpolation polynomials to|x|α(1<α<2) based on equidistant nodes in [-1,1]. Chapter four:this chapter discusses about the approximation of|χ|α on [-1,1] by interpolation polynomials. M.Revers has given an explicit construction of certain interpolating polynomials on|χ|α, we will construct a new and still simple approximation polynomial based on Chebyshev nodes and the result obtained is better than Bernstein’s and Mevers’ result.