Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class.
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 r...
| Main Authors: | , , , |
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
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2012
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| Online Access: | http://hdl.handle.net/10835/1640 |
| _version_ | 1789406363448770560 |
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| author | Beckermann, B. Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Wielonsky, F. |
| author_facet | Beckermann, B. Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Wielonsky, F. |
| author_sort | Beckermann, B. |
| collection | DSpace |
| description | 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. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-1640 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-16402023-04-12T19:37:02Z Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. Beckermann, B. Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Wielonsky, F. Análisis asintótico Polinomios ortogonales Variables aleatorias 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. 2012-08-03T09:42:29Z 2012-08-03T09:42:29Z 2004 info:eu-repo/semantics/article 1089-7658 http://hdl.handle.net/10835/1640 en info:eu-repo/semantics/openAccess Journal of Mathematical Physics 45 (11), 4239-4254 (2004) |
| spellingShingle | Análisis asintótico Polinomios ortogonales Variables aleatorias Beckermann, B. Martínez-Finkelshtein, Andrei Rakhmanov, Evgenii A. Wielonsky, F. Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. |
| title | Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. |
| title_full | Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. |
| title_fullStr | Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. |
| title_full_unstemmed | Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. |
| title_short | Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. |
| title_sort | asymptotic upper bounds for the entropy of orthogonal polynomials in the szegő class. |
| topic | Análisis asintótico Polinomios ortogonales Variables aleatorias |
| url | http://hdl.handle.net/10835/1640 |
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