Hermite Interpolation With Dickson Polynomials and Bernstein Basis Polynomials

Abstract

In this manuscript we introduce three new algorithms: (1) An algorithm to recover an unknown polynomial in terms of Dickson polynomials of the first kind, (2) an algorithm to recover an unknown polynomial in terms Dickson polynomials of the second kind, (3) an algorithm to recover an unknown polynomial in terms of Bernstein basis polynomials, from given black boxes for the polynomial itself and its first derivative. In each algorithm, we assume that the unknown polynomial has a sparse representation in the corresponding basis. The methods presented use transformations from Dickson polynomials to Laurent polynomials, a transformation from Bernstein basis polynomials to Laurent polynomials, and a recently developed algorithm as a middle step.