Asymptotic Properties of Balanced Extremal Sobolev Polynomials: Coherent Case Original Research Article

For eachn∈N andλn⩾0,Qn, λnis the monic polynomial of degreenthat minimizes the norm ‖q‖2=∫ |q|2 dμ0+λn ∫ |q′|2 dμ1in the class of all monic polynomials of degreen. Asymptotic properties of {Qn, λn} asn→∞ 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 limn n2λn=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 ofμ0andμ1; we conjecture that this result is valid in a more general setting.