Sobolev Orthogonal Polynomials on the Sierpinski Gasket

Qingxuan Jiang, Tian Lan, Kasso Okoudjou, Robert Strichartz, Shashank Sule, Sreeram Venkat, Xiaoduo Wang

Published 2020-09-30Version 1

We develop a theory of Sobolev orthogonal polynomials on the Sierpi\'nski gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $L^2$, $L^\infty$ and Sobolev norms, and study their asymptotic behaviour. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.