Solution of Interpolation Problems via the Hankel Polynomial Construction

TL;DR

This paper introduces a Hankel polynomial-based method for solving interpolation problems for polynomials and rational functions, with applications in error correction and polynomial resultant evaluation.

Contribution

It develops a novel approach using Hankel polynomials for interpolation, error correction, and resultant computation, extending classical methods by Jacobi.

Findings

Effective reconstruction of polynomials from erroneous data

New formulas for polynomial and rational interpolation

Application to polynomial resultant evaluation

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.