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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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