Sobolev Orthogonal Polynomials on the Sierpinski Gasket

Abstract

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ( $$SG$$ ), which is a fractal set that can be viewed as a limit of a sequence of finite graphs. 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 behavior. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.