Orthogonal Polynomials on the Sierpinski Gasket

Abstract

Abstract. The construction of a Laplacian on a class of fractals which includes the Sierpinski gasket (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 threeterm 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.

Sobolev Orthogonal Polynomials on the Sierpinski Gasket

“…The following lemma is motivated by [10,Theorem 3.5] and allows us to recursively compute the polynomial f n of the previous lemma. Lemma 3.2.…”

“…We conclude this introduction by pointing out that along with [10], this paper can be viewed as not only laying the foundation of a general theory of OPs on fractals, but also initiate some applications of such a theory.…”

“…In particular, there is no analog of the Weierstrass Theorem on SG, that is the set of polynomials on SG is not complete on L 2 (SG) [5,Theorem 4.3.6]. On the other hand, an initial theory of orthogonal polynomials has been developed on SG and resulted in an analog of the Legendre orthogonal polynomials on [−1, 1], [10]. More specifically, the Legendre OPs on SG arise in solving the least squares problem arg min{ f − g 2 g polynomial of order j}, where f ∈ L 2 (SG).…”

“…Some of these properties are common to the three families, while others are different. For example, all three families of OPs satisfy a three-term recurrence relation and a three-term differential equation involving the Legendre Orthogonal Polynomials on SG studied in [10].…”

“…Furthermore, by combining the three-term recurrence and the computational procedures developed in [10], we graph several SOPs not only on SG, but also on its edges. The visualization of these polynomials allows a detailed study of their qualitative properties, such as the number and location of zeroes.…”