MATH 590: Meshfree Methods - Chapter 6 - Applied Mathematics

Interpolation with Multivariate PolynomialsFor each i = 1, . . . , m, let the equation of the line L i beα i x + β i y = γ i ,where x = (x, y) ∈ R 2 .Suppose now that p interpolates zero data at all the points u i as statedabove.Since p reduces to a univariate polynomial of degree m on L m whichvanishes at m + 1 distinct points on L m , it follows that p vanishesidentically on L m , and sop(x, y) = (α m x + β m y − γ m )q(x, y),where q is a polynomial of degree m − 1.But now q satisfies the hypothesis of the theorem with m replaced bym − 1 and U replaced by Ũ consisting of the first ( )m+12 points of U.By induction, therefore q ≡ 0, and thus p ≡ 0. This establishes theuniqueness of the interpolation polynomial.fasshauer@iit.edu MATH 590 – Chapter 6 9