Interpolation of Polynomials

Abstract

Corollary 15.10 assures that there is at most one polynomial of degree n and assuming preassigned values in n + 1 given complex numbers. What we still do not know is whether such a polynomial actually exists. For instance, does there exists a polynomial f with rational coefficients, degree 3 and such that f(0) = 1, f(1) = 2, f(2) = 3 and f(3) = 0? In this chapter we study a bunch of techniques that allow us to answer this and alike questions, and which are generically referred to as interpolation of polynomials. In particular, we shall study in detail the class of Lagrange interpolating polynomials, which will then be used to solve Vandermonde’ linear systems with no Linear Algebra. In turn, the knowledge of the solutions of such linear systems will allow us to study, in Sect. 21.1, an important particular class of linear recurrence relations, thus partially extending the methods of Section 3.2 of [8].