On Bivariate Fundamental Polynomials
Vahagn Vardanyan *
Department of Mathematics and Mechanics, Yerevan State University, Armenia.
*Author to whom correspondence should be addressed.
Abstract
An n-independent set in two dimensions is a set of nodes admitting (not necessary unique) bivariate interpolation with polynomials of total degree at most n: For an arbitrary n-independent node set X we are interested with the property that each node possesses an n-fundamental polynomial in form of products of linear or quadratic factors. In the present paper we show that each node of X has an n-fundamental polynomial, which is a product of lines, if #X ≤ 2n + 1: Next we prove that each node of X has an n-fundamental polynomial, which is a product of lines or conics, if #X ≤ 2n+[n/2]+1. We bring a counterexample in each case to show that the results are not valid in general if #X ≥ 2n + 2 and #X ≥ 2n + [n/2] + 2; respectively. At the end we bring an algorithm for the construction of above mentioned fundamental polynomials. This, in view of the Lagrange formula, can be used to obtain readily also the interpolation polynomials.
Keywords: Bivariate polynomial, interpolation, fundamental polynomial, conic, n-poised nodes, n-independent nodes