Non-uniqueness of rational best approximants

Let f be a Markov function with defining measure μ supported on (−1,1), i.e., f(z)=∫(t−z)−1 dμ(t), μ⩾0, and supp(μ)⊆ ( −1,1). The uniqueness of rational best approximants to the function f in the norm of the real Hardy space H2(V), V ≔ C̄⧹D̄={z∈C̄ | |z|>1}, is investigated. It is shown that there exist Markov functions f with rational best approximants that are not unique for infinitely many numerator and denominator degrees n−1 and n, respectively. In the counterexamples, which have been constructed, the defining measures μ are rather rough. But there also exist Markov functions f with smooth defining measures μ such that the rational best approximants to f are not unique for odd denominator degrees up to a given one.