Alternation points and bivariate Lagrange interpolation

摘要

Given m+1 strictly decreasing numbers h0,h1,…,hm, we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p0,p1,…,pm such that p0=1, pj has degree j and pm−j(hn)=(−1)npj(hn) for all j,n=0,1,…,m. Using these polynomials, we construct bivariate Lagrange polynomials and cubature formulas for nodes that are points in R2 where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same (or opposite) parity.