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^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.