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Sobolev orthogonal polynomials: Balance and asymptotics. (English) Zbl 1158.33005
The authors study the asymptotic of the varying Sobolev orthogonal polynomials \(S_{n,\lambda_n}\) with respect to the inner product \[ \langle P,Q\rangle = \int PQd\mu_0 + \lambda_n \int P'Q'd\mu_1,\quad \lambda_0>0, \] where \(\mu_0\) and \(\mu_1\) are measures supported on an unbounded interval. Here they focus on the balanced case which implies, in general, that \(\lambda_n\) in a non constant sequence.
In the first part of the paper they answer the question: What is the appropriate choice of the sequence \((\lambda_n)\)? The answer is, instead of the natural choice \(\lambda_n n^2\sim \text{const.}\), the value \(\lambda_n n^2\sim a_{n+1}\), where \(a_n\) is the so called Mhaskar-Rakhmanov-Saff number associated with the weight \(w(z)=\exp(-Q(z))\), for a certain function \(Q\).
In the second part of the paper they apply these ideas developed in the first part for obtaining the asymptotic for the Freud weight \(d\mu_0=d\mu_1=\exp(-x^4)\). The paper is rather well written and the exposition is quite clear which add more value to the interesting results obtained by the authors.

MSC:
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
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