Shannon entropy of symmetric Pollaczek polynomials
We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$ E_n= - \int_{-1}^1 \log \big(p^2_n(x) \big) p^2_n(x) w(x) d x, $$ and $$ F_n = -\int_{-1}^1 \log (p_n^2(x)w(x)) p_n^2(x) w(x) dx, $$ when $w$ is the symmetric Pollaczek weight on $[-1,1]$ with main parameter $\lambda...
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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/10835/1635 |
| _version_ | 1789406473235726336 |
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| author | Martínez-Finkelshtein, Andrei Sánchez-Lara, J. F. |
| author_facet | Martínez-Finkelshtein, Andrei Sánchez-Lara, J. F. |
| author_sort | Martínez-Finkelshtein, Andrei |
| collection | DSpace |
| description | We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$ E_n= - \int_{-1}^1 \log \big(p^2_n(x) \big) p^2_n(x) w(x) d x, $$ and $$ F_n = -\int_{-1}^1 \log (p_n^2(x)w(x)) p_n^2(x) w(x) dx, $$ when $w$ is the symmetric Pollaczek weight on $[-1,1]$ with main parameter $\lambda\geq 1$, and $p_n$ is the corresponding orthonormal polynomial of degree $n$. It is well known that $w$ does not belong to the Szeg\H{o} class, which implies in particular that $E_n\to -\infty$. For this sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that $F_n \to \log (\pi)-1$, proving that this ``universal behavior'' extends beyond the Szeg\H{o} class. The asymptotics of $E_n$ has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with $p_n$'s. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-1635 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-16352023-04-12T19:37:40Z Shannon entropy of symmetric Pollaczek polynomials Martínez-Finkelshtein, Andrei Sánchez-Lara, J. F. Polinomios simétricos Pollaczek Entropía de Shannon Comportamiento asintótico Integrales entrópicas Shannon entropy Symmetric Pollaczek polynomials Asymptotic behavior Entropic integrals We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$ E_n= - \int_{-1}^1 \log \big(p^2_n(x) \big) p^2_n(x) w(x) d x, $$ and $$ F_n = -\int_{-1}^1 \log (p_n^2(x)w(x)) p_n^2(x) w(x) dx, $$ when $w$ is the symmetric Pollaczek weight on $[-1,1]$ with main parameter $\lambda\geq 1$, and $p_n$ is the corresponding orthonormal polynomial of degree $n$. It is well known that $w$ does not belong to the Szeg\H{o} class, which implies in particular that $E_n\to -\infty$. For this sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that $F_n \to \log (\pi)-1$, proving that this ``universal behavior'' extends beyond the Szeg\H{o} class. The asymptotics of $E_n$ has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with $p_n$'s. 2012-08-03T08:25:59Z 2012-08-03T08:25:59Z 2007 info:eu-repo/semantics/article http://hdl.handle.net/10835/1635 en info:eu-repo/semantics/openAccess Journal of Approximation Theory Vol. 145 Nº 1 (2007) |
| spellingShingle | Polinomios simétricos Pollaczek Entropía de Shannon Comportamiento asintótico Integrales entrópicas Shannon entropy Symmetric Pollaczek polynomials Asymptotic behavior Entropic integrals Martínez-Finkelshtein, Andrei Sánchez-Lara, J. F. Shannon entropy of symmetric Pollaczek polynomials |
| title | Shannon entropy of symmetric Pollaczek polynomials |
| title_full | Shannon entropy of symmetric Pollaczek polynomials |
| title_fullStr | Shannon entropy of symmetric Pollaczek polynomials |
| title_full_unstemmed | Shannon entropy of symmetric Pollaczek polynomials |
| title_short | Shannon entropy of symmetric Pollaczek polynomials |
| title_sort | shannon entropy of symmetric pollaczek polynomials |
| topic | Polinomios simétricos Pollaczek Entropía de Shannon Comportamiento asintótico Integrales entrópicas Shannon entropy Symmetric Pollaczek polynomials Asymptotic behavior Entropic integrals |
| url | http://hdl.handle.net/10835/1635 |
| work_keys_str_mv | AT martinezfinkelshteinandrei shannonentropyofsymmetricpollaczekpolynomials AT sanchezlarajf shannonentropyofsymmetricpollaczekpolynomials |