Constructive Approximation

, Volume 30, Issue 1, pp 93–119 | Cite as

Discrete Entropies of Orthogonal Polynomials

  • A. I. Aptekarev
  • J. S. Dehesa
  • A. Martínez-Finkelshtein
  • R. Yáñez


Given a nontrivial Borel measure on ℝ, let p n be the corresponding orthonormal polynomial of degree n whose zeros are λ j (n) , j=1,…,n. Then for each j=1,…,n,
$$\vec{\Psi}_{j}^{2}\stackrel {\mathrm {def}}{=}\bigl(\Psi_{1j}^{2},\dots,\Psi_{nj}^{2}\bigr)$$
$$\Psi_{ij}^{2}=p_{i-1}^{2}\bigl(\lambda_{j}^{(n)}\bigr)\Biggl(\sum_{k=0}^{n-1}p_{k}^{2}\bigl(\lambda_{j}^{(n)}\bigr)\Biggr)^{-1},\quad i=1,\dots,n,$$
defines a discrete probability distribution. The Shannon entropy of the sequence {p n } is consequently defined as
$$\mathcal{S}_{n,j}\stackrel {\mathrm {def}}{=}-\sum_{i=1}^{n}\Psi_{ij}^{2}\log\bigl(\Psi_{ij}^{2}\bigr).$$
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 number theory. In addition, several results of numerical computations exemplifying the behavior of \(\mathcal{S}_{n,j}\) for other families are presented.


Orthogonal polynomials Shannon entropy Chebyshev polynomials Euler–Maclaurin formula 

AMS Subject Classifications

33C45 41A58 42C05 94A17 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • A. I. Aptekarev
    • 1
  • J. S. Dehesa
    • 2
  • A. Martínez-Finkelshtein
    • 3
    • 2
  • R. Yáñez
    • 2
  1. 1.Keldysh Institute of Applied MathematicsMoscowRussia
  2. 2.Institute Carlos I of Theoretical and Computational PhysicsGranada UniversityGranadaSpain
  3. 3.Department of Statistics and Applied MathematicsUniversity of AlmeríaAlmeríaSpain

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