2015/05/04 by Vardanyan, Vahagn
#14H50 #41A05 #41A63 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1505.00574
An n-independent set in two dimensions is a set of nodes admitting (not necessarily unique) bivariate interpolation with polynomials of total degree at most n. For an arbitrary n-independent node set \mathcal X we are interested with the property that each node possesses an n-fundamental polynomial in form of product of linear or quadratic factors. In the present paper we show that each node of \mathcal X has an n-fundamental polynomial, which is a product of lines, if #\mathcal X≤ 2n+1. Next we prove that each node of \mathcal X has an n-fundamental polynomial, which is a product of lines or conics, if #\mathcal X≤ 2n+[n/2]+1. We have a counterexample in each case to show that the results are not valid in general if #\mathcal X≥ 2n+2 and #\mathcal X≥ 2n+[n/2]+2, respectively.