Orthogonal polynomials of equilibrium measures supported on Cantor sets

Abstract

We study the orthogonal polynomials associated with the equilibrium measure, in logarithmic potential theory, living on the attractor of an Iterated Function System. We construct sequences of discrete measures, that converge weakly to the equilibrium measure, and we compute their Jacobi matrices via standard procedures, suitably enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, that show stability and efficiency of the whole procedure. As a secondary result, we also compute Jacobi matrices of equilibrium measures on finite sets of intervals, and of balanced measures of Iterated Function Systems.These algorithms can reach large orders: we study the asymptotic behavior of the orthogonal polynomials and we show that they can be used to efficiently compute Green's functions and conformal mappings of interest in constructive function theory.

Minkowski's Question Mark Measure

“…This structure is also hierarchically encoded in the Jacobi matrix of the measure, in the sense that the finer structure the higher the index of the Jacobi matrix entries that reproduce it. This happens much in the same way as what observed in [39] for the Jacobi matrix -log(w 60000 ) -Λ 60000 log(K 60000 ) -log(w 30000 ) -Λ 30000 log(K 30000 ) -log(w 15000 ) -Λ 15000 log(K 15000 ) -log(w 7500 ) -Λ 7500 log(K 7500 ) ). Geometrically increasing values of j are plotted.…”

“…To compute the Jacobi matrix of the discrete measure η n , one can use the discretized Stieltjes algorithm [17], or better Gragg and Harrod's algorithm [27], which is more stable and can be enhanced to treat large sets of atoms [39]. Nonetheless, this approach is only helpful if relatively small values of n are sufficient to yield a significant number of Jacobi matrix entries (at convergence), because of the geometrical increase with n of the computational complexity.…”

Asymptotic Properties of Jacobi Matrices for a Family of Fractal Measures

“…One can compare these values with Fig. 2 in [39]. It was shown (for the stretched version of this set but similar arguments are valid for this case also) in [5] that K(γ) is a generalized polynomial Julia set (see e.g.…”

“…There are many open problems regarding orthogonal polynomials on Cantor sets, such as how to define the Szegő class of measures and isospectral torus (see e.g. [21,22] for the previous results and [32,33,36,38,39] for possible extensions of the theory and important conjectures) especially when the support has zero Lebesgue measure. The family of sets that we consider here contains both positive and zero Lebesgue measure sets, Parreau-Widom and non Parreau-Widom sets.…”