Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class

Details Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class

2003-11-05

arxiv.org/abs/math/0311055v1

Mathematics

Classical Analysis and ODEs

17 pages

Scientific paper

10.1063/1.1794842

We give an asymptotic upper bound as $n\to\infty$ for the entropy integral $$E_n(w)= -\int p_n^2(x)\log (p_n^2(x))w(x)dx,$$ where $p_n$ is the $n$th degree orthonormal polynomial with respect to a weight $w(x)$ on $[-1,1]$ which belongs to the Szeg\H{o} class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights $w$ in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when $w(x)$ belongs to the Szeg\H{o} class. In each case, we give conditions for these upper bounds to be attained.

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