Asymptotic properties of balanced extremal Sobolev polynomials: Coherent case

For each n ∈ ℕ and λn ≥ 0, Qn, λn is the monic polynomial of degree n that minimizes the norm ∥ q ∥ 2 = ∫ \q\2 dμ0 + λn ∫ \q́\ 2 dμ1 in the class of all monic polynomials of degree n. Asymptotic properties of {Qn, λn} as n → ∞ are studied under additional assumption that (μ0, μ1) is a coherent pair of measures on [ - 1, 1] and the sequence {λn} is regularly decreasing and satisfies lim n n2 λn = L ∈ [ 0, + ∞]. The behavior of the norms and zeros of these polynomials is also studied. We show that in some cases the sequence {Qn, λn} asymptotically behaves as the monic orthogonal polynomials sequence corresponding to a new measure constructed as a combination of μ0 and μ1 ; we conjecture that this result is valid in a more general setting.