Sobolev Orthogonal Polynomials on the Sierpinski Gasket

2020/09/30 by Jiang, Qingxuan, Lan, Tian, Okoudjou, Kasso +4
#28A80 #33A99 #33F05 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.00107

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 L², L^∞ 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.

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