Sobolev Orthogonal Polynomials on the Sierpinski Gasket

TL;DR

This paper develops Sobolev orthogonal polynomials on the Sierpinski gasket, establishing recurrence relations, norm estimates, asymptotic behavior, and applications to quadrature and interpolation.

Contribution

It introduces a novel theory of Sobolev orthogonal polynomials on fractals, including recurrence relations and computational tools for applications.

Findings

Recurrence relations for Sobolev orthogonal polynomials on $SG$

Norm estimates in various function spaces

Development of fast computational methods for applications

Abstract

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.