Lecture Details
Name: Larry Harris (Kentucky)Speaking: Wed 8th May 14:30 - 14:55Title: Bivariate interpolation and cubature on the even-odd product nodesAbstract: Let $h_0 > h_1 > \cdots > h_m$ and $\tilde{h}_0 > \tilde{h}_1 > \cdots >\tilde{h}_m$ be arbitrary numbers. The even (resp., odd) product nodes are those points $(h_n,\tilde{h}_q)$ where $n$ and $q$ have the same (resp., opposite) parity. There is an algorithm producing orthogonal polynomials $p_0,p_1,\ldots,p_m$ such that \[ p_{m-j}(h_n) = (-1)^n p_j(h_n), \quad 0\leq n,j\leq m, \] and a similar set of orthogonal polynomials for $\tilde{h}_0 , \tilde{h}_1, \ldots, \tilde{h}_m$. Using these polynomials and the corresponding two-dimensional reproducing kernel, we construct Lagrange polynomials of degree $m$ for each of the two sets of product nodes. This form of Lagrange polynomials leads to a cubature theorem that expresses a double integral of a bivariate polynomial of degreee $2m-1$ as a weighted sum of the values of the polynomial on the even or odd nodes.