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 L2, L∞, 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.