Orthogonal Polynomials on Bubble-Diamond Fractals

TL;DR

This paper develops a theory of orthogonal polynomials on bubble-diamond fractals, extending classical polynomial concepts to fractal sets and establishing recursion relations for these polynomials.

Contribution

It introduces a new framework for polynomials on fractals, specifically constructing orthogonal polynomials and proving their three-term recursion formula.

Findings

Established a polynomial theory on bubble-diamond fractals

Constructed orthogonal polynomials as multiharmonic functions

Proved the three-term recursion formula for these polynomials

Abstract

We develop a theory of polynomials and, in particular, an analog of the theory of Legendre orthogonal polynomials on the bubble-diamond fractals, a class of fractal sets that can be viewed as the completion of a limit of a sequence of finite graph approximations. In this setting, a polynomial of degree $j$ can be viewed as a multiharmonic function, a solution of the equation $\Delta^{j+1}u=0$. We prove that the sequence of orthogonal polynomials we construct obeys a three-term recursion formula.