Orthogonal Polynomials on Bubble-Diamond Fractals

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.