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In the paper "Infinite product representations for kernels and iterations offunctions", the authors associate certain Fatou subsets with reproducing kernelHilbert spaces. They also present a method for constructing an orthonormalbasis for said Hilbert space, but the method depends on the polynomial of thegiven Fatou set. We provide a partial classification of those polynomials themethod applies to.

Partial Classification of Polynomials and an Orthonormal Basis Construction on the Associated Basin of Attraction