Sobolev Orthogonal Polynomials on the Sierpinski Gasket

Abstract

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński 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.

References in corpus (1)

• Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series