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Discrete entropies of orthogonal polynomials
dc.contributor.author | Aptekarev, A. I. | |
dc.contributor.author | Dehesa, J. S. | |
dc.contributor.author | Martínez Finkelshtein, Andrei | |
dc.contributor.author | Yáñez, R. | |
dc.date.accessioned | 2012-08-01T08:45:54Z | |
dc.date.available | 2012-08-01T08:45:54Z | |
dc.date.issued | 2009 | |
dc.identifier.uri | http://hdl.handle.net/10835/1630 | |
dc.description.abstract | 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. | es_ES |
dc.language.iso | en | es_ES |
dc.source | Constructive Approximation Vol. 30 Nº 1 (2009) | es_ES |
dc.subject | Polinomios ortogonales | es_ES |
dc.subject | Entropía de Shannon | es_ES |
dc.subject | Chebyshev polinomios | es_ES |
dc.subject | Fórmula Euler–Maclaurin | es_ES |
dc.subject | Orthogonal polynomials | es_ES |
dc.subject | Shannon entropy | es_ES |
dc.subject | Chebyshev polynomials | es_ES |
dc.subject | Euler–Maclaurin formula | es_ES |
dc.title | Discrete entropies of orthogonal polynomials | es_ES |
dc.type | info:eu-repo/semantics/article | es_ES |
dc.rights.accessRights | info:eu-repo/semantics/openAccess | es_ES |