Solution of Interpolation Problems via the Hankel Polynomial Construction

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 $\{\sum_{j=1}^N x_j^ky_j/W^{\prime}(x_j) \}_{k\in \mathbb N}$ and $\{\sum_{j=1}^N x_j^k/(y_jW^{\prime}(x_j)) \}_{k\in \mathbb N}$; 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.