  In the paper "Infinite Product Represenations for Kernels and Iterations of
Functions", a technique was developed which allows for the construction of a
reproducing kernel Hilbert space on basins of attraction containing $0$. When
the right conditions are met, an explicit orthonormal basis can be constructed
using a particular class of operators. It is natural then to consider how the
orthonormal basis changes as we let the basin of attraction vary. We will
consider this question for the basins of attraction containing $0$ of the
family of polynomials $\mathcal{F} = \{az^{2^{n+2}}-2az^{2^{n+1}}:a\neq0\}$,
where $n\in\mathbb{N}$.
