Discrete entropies of orthogonal polynomials
Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $\lambda_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec \Psi_j^2 = (\Psi_{1j}^2, ..., \Psi_{nj}^2) $$ with $$ \Psi_{ij}^2= p_{i-1}^2 (\lambda_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(\lambda_j^{(n)}))^{-1}, \quad...
| Main Authors: | , , , |
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
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2012
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| Online Access: | http://hdl.handle.net/10835/1630 |
| _version_ | 1789406449782226944 |
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| author | Aptekarev, A. I. Dehesa, J. S. Martínez-Finkelshtein, Andrei Yáñez, R. |
| author_facet | Aptekarev, A. I. Dehesa, J. S. Martínez-Finkelshtein, Andrei Yáñez, R. |
| author_sort | Aptekarev, A. I. |
| collection | DSpace |
| description | Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $\lambda_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec \Psi_j^2 = (\Psi_{1j}^2, ..., \Psi_{nj}^2) $$ with $$ \Psi_{ij}^2= p_{i-1}^2 (\lambda_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(\lambda_j^{(n)}))^{-1}, \quad i=1, >..., n, $$ defines a discrete probability distribution. The Shannon entropy of the sequence $\{p_n\}$ is consequently defined as $$ \mathcal S_{n,j} = -\sum_{i=1}^n \Psi_{ij}^{2} \log (\Psi_{ij}^{2}) . $$ In the case of Chebyshev polynomials of the first and second kinds an explicit and closed formula for $\mathcal S_{n,j}$ is obtained, revealing interesting connections with the number theory. Besides, several results of numerical computations exemplifying the behavior of $\mathcal S_{n,j}$ for other families are also presented. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-1630 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-16302023-04-12T19:37:30Z Discrete entropies of orthogonal polynomials Aptekarev, A. I. Dehesa, J. S. Martínez-Finkelshtein, Andrei Yáñez, R. Polinomios ortogonales Entropía de Shannon Chebyshev polinomios Fórmula Euler–Maclaurin Orthogonal polynomials Shannon entropy Chebyshev polynomials Euler–Maclaurin formula Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $\lambda_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec \Psi_j^2 = (\Psi_{1j}^2, ..., \Psi_{nj}^2) $$ with $$ \Psi_{ij}^2= p_{i-1}^2 (\lambda_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(\lambda_j^{(n)}))^{-1}, \quad i=1, >..., n, $$ defines a discrete probability distribution. The Shannon entropy of the sequence $\{p_n\}$ is consequently defined as $$ \mathcal S_{n,j} = -\sum_{i=1}^n \Psi_{ij}^{2} \log (\Psi_{ij}^{2}) . $$ In the case of Chebyshev polynomials of the first and second kinds an explicit and closed formula for $\mathcal S_{n,j}$ is obtained, revealing interesting connections with the number theory. Besides, several results of numerical computations exemplifying the behavior of $\mathcal S_{n,j}$ for other families are also presented. 2012-08-01T08:45:54Z 2012-08-01T08:45:54Z 2009 info:eu-repo/semantics/article http://hdl.handle.net/10835/1630 en info:eu-repo/semantics/openAccess Constructive Approximation Vol. 30 Nº 1 (2009) |
| spellingShingle | Polinomios ortogonales Entropía de Shannon Chebyshev polinomios Fórmula Euler–Maclaurin Orthogonal polynomials Shannon entropy Chebyshev polynomials Euler–Maclaurin formula Aptekarev, A. I. Dehesa, J. S. Martínez-Finkelshtein, Andrei Yáñez, R. Discrete entropies of orthogonal polynomials |
| title | Discrete entropies of orthogonal polynomials |
| title_full | Discrete entropies of orthogonal polynomials |
| title_fullStr | Discrete entropies of orthogonal polynomials |
| title_full_unstemmed | Discrete entropies of orthogonal polynomials |
| title_short | Discrete entropies of orthogonal polynomials |
| title_sort | discrete entropies of orthogonal polynomials |
| topic | Polinomios ortogonales Entropía de Shannon Chebyshev polinomios Fórmula Euler–Maclaurin Orthogonal polynomials Shannon entropy Chebyshev polynomials Euler–Maclaurin formula |
| url | http://hdl.handle.net/10835/1630 |
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