Bivariate polynomial interpolation on the square at new nodal sets
Let \(\Omega\) be a compact from \({\mathbb R}^d\). Set \(X_N^*\) from \(\Omega\) of cardinality \(N\) is called an optimal polynomial interpolation points, if \(\Lambda_n(X_N)=\max_{x\in\Omega}\sum_{i=1}^N| l_i({\mathbf x},X_N)| \) attains it's minimum on \(X_N^*\). Here \(X_N=\{{\mathbf x}_1,\ldots,{\mathbf x}_n\}\) is contained in \(\Omega\), \(l_i\) are fundamental Lagrange polynomials of degree \(n\), \(N=(n+d)!/n!d!\). The article is devoted to applied questions. The authors experiment with four families of points, which are asymptotically equidistributed with respect to the Dubiner metric and estimate growth of the Lebesgue constants \(\Lambda_n(X_N)\). So called Padua points on \([-1,1]^2\) give experimental relation \(\Lambda_n=O(\log^2n)\). There are no proofs in the paper.
scientific article; zbMATH DE number 2175488
scientific article