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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.

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
Full Text: DOI
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