Orthogonal Polynomials on the Sierpinski Gasket
arXiv:1110.1554
Abstract
The construction of a Laplacian on a class of fractals which includes the Sierpinski gasket ({\bf }) has given rise to an intensive research on analysis on fractals. For instance, a complete theory of polynomials and power series on 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 . In particular, we investigate key properties of these OP on , 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.
29 pages, 10 figures, 2 tables. Include short version of previous section 5 in subsection 4.4; correct typos, reduce the number of figures