Advertisement

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
Article

Abstract

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,
$$\vec{\Psi}_{j}^{2}\stackrel {\mathrm {def}}{=}\bigl(\Psi_{1j}^{2},\dots,\Psi_{nj}^{2}\bigr)$$
with
$$\Psi_{ij}^{2}=p_{i-1}^{2}\bigl(\lambda_{j}^{(n)}\bigr)\Biggl(\sum_{k=0}^{n-1}p_{k}^{2}\bigl(\lambda_{j}^{(n)}\bigr)\Biggr)^{-1},\quad i=1,\dots,n,$$
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.

Keywords

Orthogonal polynomials Shannon entropy Chebyshev polynomials Euler–Maclaurin formula 

AMS Subject Classifications

33C45 41A58 42C05 94A17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972) zbMATHGoogle Scholar
  2. 2.
    Anderson, E., Bai, Z., Bischof, C.H., Blackford, S., Demmel, J., Dongarra, J.J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.C.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999). http://www.netlib.org/lapack/lug/ Google Scholar
  3. 3.
    Aptekarev, A.I., Buyarov, V.S., Dehesa, J.S.: Asymptotic behavior of the L p-norms and the entropy for general orthogonal polynomials. Russ. Acad. Sci. Sb. Math. 82(2), 373–395 (1995) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Aptekarev, A.I., Buyarov, V.S., Dehesa, J.S., Van Assche, W.: Asymptotics for entropy integrals of orthogonal polynomials. Russ. Acad. Sci. Dokl. Math. 53, 47–49 (1996) zbMATHGoogle Scholar
  5. 5.
    Babenko, K.I.: An inequality in the theory of Fourier integrals. Izv. Akad. Nauk SSSR Ser. Mat. 25, 531–542 (1961). Engl. transl.: Am. Math. Soc., Transl., II. Ser. 44, 115–128 (1965) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Barker, V.A., Blackford, S., Dongarra, J.J., Du Croz, J., Hammarling, S., Marinova, M., Wa’sniewski, J., Yalamov, P.: LAPACK95 Users’ Guide. SIAM, Philadelphia (2001). www.netlib.org/lapack95/lug95/ Google Scholar
  7. 7.
    Beckermann, B., Martínez-Finkelshtein, A., Rakhmanov, E.A., Wielonsky, F.: Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class. J. Math. Phys. 45(11), 4239–4254 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bialynicki-Birula, I.: Entropic uncertainty relations. Phys. Lett. A 103, 253–254 (1984) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bialynicki-Birula, I., Mycielsky, J.: Uncertainty relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129–132 (1975) CrossRefGoogle Scholar
  10. 10.
    Buyarov, V., Dehesa, J.S., Martínez-Finkelshtein, A., Sánchez-Lara, J.: Computation of the entropy of polynomials orthogonal on an interval. SIAM J. Sci. Comput. 26(2), 488–509 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Buyarov, V.S., Dehesa, J.S., Martínez-Finkelshtein, A., Saff, E.B.: Asymptotics of the information entropy for Jacobi and Laguerre polynomials with varying weights. J. Approx. Theory 99(1), 153–166 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Buyarov, V.S., López-Artés, P., Martínez-Finkelshtein, A., Van Assche, W.: Information entropy of Gegenbauer polynomials. J. Phys. A 33(37), 6549–6560 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dehesa, J.S., Martínez-Finkelshtein, A., Sánchez-Ruiz, J.: Quantum information entropies and orthogonal polynomials. In: Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications, Patras, 1999. J. Comput. Appl. Math. 133, 23–46 (2001) Google Scholar
  14. 14.
    Dreizler, R.M., Gross, E.K.U.: Density Functional Theory: An Approach to the Quantum Mechanics. Springer, Heidelberg (1990) Google Scholar
  15. 15.
    Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. B 136, 864–870 (1964) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications, vol. 98. Cambridge University Press, Cambridge (2005). With two chapters by Walter Van Assche, with a foreword by Richard A. Askey zbMATHGoogle Scholar
  17. 17.
    Jacquet, P., Szpankowski, W.: Entropy computations via analytic de-Poissonization. IEEE Trans. Inform. Theory 45(4), 1072–1081 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Knessl, C.: Integral representations and asymptotic expansions for Shannon and Renyi entropies. Appl. Math. Lett. 11(2), 69–74 (1998) CrossRefMathSciNetGoogle Scholar
  19. 19.
    March, N.H.: Electron Density Theory of Atoms and Molecules. Academic Press, New York (1992) Google Scholar
  20. 20.
    Martínez-Finkelshtein, A., Sánchez-Lara, J.F.: Shannon entropy of symmetric Pollaczek polynomials. J. Approx. Theory 145(1), 55–80 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rutter, J.: A serial implementation of Cuppen’s divide and conquer algorithm for the symmetric eigenvalue problem. Technical Report CS-94-225, Department of Computer Science, University of Tennessee, Knoxville, TN, USA, March 1994. LAPACK Working Note 69 Google Scholar
  22. 22.
    Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, New York (1993) zbMATHGoogle Scholar
  23. 23.
    Sidi, A.: Euler-Maclaurin expansions for integrals with endpoint singularities: a new perspective. Numer. Math. 98(2), 371–387 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sondow, J., Weisstein, E.W.: Riemann zeta function. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannZetaFunction.html

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

Personalised recommendations