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    Discrete entropies of orthogonal polynomials

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    Aptekarev-Discrete entropies.pdf (278.0Kb)
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    URI: http://hdl.handle.net/10835/1630
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    Author/s
    Aptekarev, A. I.; Dehesa, J. S.; Martínez-Finkelshtein, Andrei; Yáñez, R.
    Date
    2009
    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.
    Palabra/s clave
    Polinomios ortogonales
    Entropía de Shannon
    Chebyshev polinomios
    Fórmula Euler–Maclaurin
    Orthogonal polynomials
    Shannon entropy
    Chebyshev polynomials
    Euler–Maclaurin formula
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    • Artículos de revista Dpto. Matemáticas [119]

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