Solution of Interpolation Problems via the Hankel Polynomial Construction

Abstract

We treat the interpolation problem $\{f(x_j)=y_j{\}}_{j=1}^N$ for polynomial and rational functions. Developing the approach by C.Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences $\{{\sum }_{j=1}^N{x}_j^k{y}_j/W^{\prime }(x_j){\}}_{k\in \mathbb{N}}$ and $\{{\sum }_{j=1}^N{x}_j^k/(y_j{W}^{\prime }(x_j)){\}}_{k\in \mathbb{N}}$; here $W(x)={\prod }_{j=1}^N(x-x_j)$. The obtained results are applied for the error correction problem, i.e. the problem of reconstructing the polynomial from a redundant set of its values some of which are probably erroneous. The problem of evaluation of the resultant of polynomials $p(x)$ and $q(x)$ from the set of values $\{p(x_j)/q(x_j){\}}_{j=1}^N$ is also tackled within the framework of this approach.

56 pages, 1 figure