arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0710.2134math.CAcs.ITmath-phmath.ITmath.MP
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正交多项式的离散熵

Discrete entropies of orthogonal polynomials

A. I. Aptekarev, J. S. Dehesa, A. Martinez-Finkelshtein, R. Yañez

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26 pages, 6 figures
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摘要：

令$p_n$为实数线上的第$n$次正交多项式，其零点为$λ_j^{(n)}$、$j=1, ..., n$。然后，对于每个$j=1, ..., n$，具有$$ Ψ_{ij}^2= p_{i-1}^2 (λ_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(λ_j^{(n)}))^{-1}, \quad i=1, >..., n, $$的$$ \vec Ψ_j^2 = (Ψ_{1j}^2, ..., Ψ_{nj}^2) $$定义离散概率分布。序列$\{p_n\}$的香农熵因此被定义为$$ \mathcal S_{n,j} = -\sum_{i=1}^n Ψ_{ij}^{2} \log (Ψ_{ij}^{2}) . $$。在第一和第二类切比雪夫多项式的情况下，得到了$\mathcal S_{n,j}$的显式和封闭公式，揭示了与数论的有趣联系。此外，还给出了几个数值计算结果，举例说明了$\mathcal S_{n,j}$对其他族的行为。

英文摘要：

Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $λ_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec Ψ_j^2 = (Ψ_{1j}^2, ..., Ψ_{nj}^2) $$ with $$ Ψ_{ij}^2= p_{i-1}^2 (λ_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(λ_j^{(n)}))^{-1}, \quad i=1, >..., n, $$ defines a discrete probability distribution. The Shannon entropy of the sequence $\{p_n\}$ is consequently defined as $$ \mathcal S_{n,j} = -\sum_{i=1}^n Ψ_{ij}^{2} \log (Ψ_{ij}^{2}) . $$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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