Orthogonal Polynomials on the Sierpinski Gasket

TL;DR

This paper develops and analyzes orthogonal polynomials on the Sierpinski gasket, including their properties, recursion relations, asymptotics, and zero sets, extending classical polynomial theory to fractal domains.

Contribution

It constructs and studies orthogonal polynomials on the Sierpinski gasket, introducing recursion formulas, asymptotic analysis, and numerical visualization methods.

Findings

Established a three-term recursion formula for OP on SG

Analyzed asymptotic behavior of recursion coefficients

Numerically visualized zero and nodal sets of the polynomials

Abstract

The construction of a Laplacian on a class of fractals which includes the Sierpinski gasket ({\bf $SG$}) has given rise to an intensive research on analysis on fractals. For instance, a complete theory of polynomials and power series on $SG$ has been developed by one of us and his coauthors. We build on this body of work to construct certain analogs of classical orthogonal polynomials (OP) on $SG$. In particular, we investigate key properties of these OP on $SG$, including a three-term recursion formula and the asymptotics of the coefficients appearing in this recursion. Moreover, we develop numerical tools that allow us to graph a number of these OP. Finally, we use these numerical tools to investigate the structure of the zero and the nodal sets of these polynomials.