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Discrete entropies of orthogonal polynomials. (English) Zbl 1177.33018
The authors introduce the discrete entropy of orthonormal polynomials defined as the Shannon entropy of the probability distribution constructed as follows. For a nontrivial Borel measure on the real line, they consider the corresponding orthonormal polynomial of degree \(n\), \(p_n\), whose zeros are \( \lambda_j^{(n)},\; j=1,\dots, n\). Then for each \(j=1,\dots, n,\) \[ \overrightarrow{\Psi}_j^2=(\Psi_{1j}^2,\dots,\Psi_{nj}^2)\;\; \text{with}\;\; \Psi_{ij}^2= p_{i-1}^2(\lambda_j^{(n)})\left(\sum _{k=0}^{n-1}p_k^2(\lambda_j^{(n)})\right)^{-1}, \] defines a discrete probability distribution. The Shannon entropy of the sequence \(\{p_n\}\) is defined as \(\mathcal{S}_{n,j}=- \sum _{i=1}^n \Psi_{ij}^2 \log (\Psi_{ij}^2).\)
Since there are no known results for the Shannon entropy of the orthogonal polynomials’ related distributions, it is important to point out that the authors compute explicitly the discrete entropy \(\mathcal{S}_{n,j}\) corresponding to the Chebyshev orthonormal polynomials of the first and second kinds and they also find the first two terms of the asymptotic expansion of \(\mathcal{S}_{n,j}\) for fixed \(j\) and large \(n\). The formulas exhibit nice connections with relevant objects from number theory.
Finally, they present some results of numerical evaluation of the entropy for several orthogonal polynomials.

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
94A17 Measures of information, entropy
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