Sobolev Orthogonal Polynomials on the Sierpinski Gasket

Overview

Building on the theory of Legendre orthogonal polynomials on the Sierpinski Gasket (SG), we develop a theory of Sobolev orthogonal polynomials on SG. Initially, we define several notions of a Sobolev inner product on SG using powers of the canonical Laplacian. We use these inner products to find general recurrence relations connecting the Sobolev polynomials to the Legendre polynomials on SG. We then analyze the finer properties of the Sobelev inner product by presenting estimates for the \(L^2\), \(L^\infty\) and \(H^m\) norms of the polynomials and studying their convergence properties with respect to the parameters in the \(H^m\) inner product. We also highlight the major differences and similarities between the polynomials on SG and those on \(\mathbb{R}\) resulting from the properties of the self-similar measure and the Laplacian. Finally, we develop fast computational tools to study nodal domains and applications to quadrature and interpolation on SG.

This website shows the theoretical and numerical results, as well as the programs.

This research was a joint work by Max Jiang, Tian Lan, Kasso Okoudjou, Robert Strichartz, Shashank Sule, Sreeram Venkat and Xiaoduo Wang at Cornell University in 2019 summer REU.