Discrete entropies of orthogonal polynomials
| dc.creator | Aptekarev, A. I. | |
| dc.creator | Dehesa, J. S. | |
| dc.creator | Martinez-Finkelshtein, A. | |
| dc.creator | Yañez, R. | |
| dc.date | 2007-10-10 | |
| dc.date.accessioned | 2026-07-07T08:35:39Z | |
| dc.date.available | 2026-07-07T08:35:39Z | |
| dc.description | 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. | |
| dc.description | 26 pages, 6 figures | |
| dc.identifier | https://arxiv.org/abs/0710.2134 | |
| dc.identifier | http://arxiv.org/abs/0710.2134 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/139814 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Information Theory | |
| dc.subject | Mathematical Physics | |
| dc.subject | 33C45; 41A58; 42C05; 94A17 | |
| dc.title | Discrete entropies of orthogonal polynomials | |
| dc.type | text |