zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972) · Zbl 0543.33001
[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/ · Zbl 0934.65030
[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) · Zbl 0862.46014 · doi:10.1070/SM1995v082n02ABEH003571
[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) · Zbl 0892.33005
[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) · Zbl 0122.34404
[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/ · Zbl 0992.65013
[7] Beckermann, B., Martínez-Finkelshtein, A., Rakhmanov, E.A., Wielonsky, F.: Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szego class. J. Math. Phys. 45(11), 4239–4254 (2004) · Zbl 1064.42014 · doi:10.1063/1.1794842
[8] Bialynicki-Birula, I.: Entropic uncertainty relations. Phys. Lett. A 103, 253–254 (1984) · doi:10.1016/0375-9601(84)90118-X
[9] Bialynicki-Birula, I., Mycielsky, J.: Uncertainty relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129–132 (1975) · doi:10.1007/BF01608825
[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) · Zbl 1082.33004 · doi:10.1137/S1064827503426711
[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) · Zbl 0946.33005 · doi:10.1006/jath.1998.3315
[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) · Zbl 1008.81015 · doi:10.1088/0305-4470/33/37/307
[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) · Zbl 1008.81014
[14] Dreizler, R.M., Gross, E.K.U.: Density Functional Theory: An Approach to the Quantum Mechanics. Springer, Heidelberg (1990) · Zbl 0723.70002
[15] Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. B 136, 864–870 (1964)
[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 · Zbl 1082.42016
[17] Jacquet, P., Szpankowski, W.: Entropy computations via analytic de-Poissonization. IEEE Trans. Inform. Theory 45(4), 1072–1081 (1999) · Zbl 0959.94009 · doi:10.1109/18.761251
[18] Knessl, C.: Integral representations and asymptotic expansions for Shannon and Renyi entropies. Appl. Math. Lett. 11(2), 69–74 (1998) · Zbl 1337.94007 · doi:10.1016/S0893-9659(98)00013-5
[19] March, N.H.: Electron Density Theory of Atoms and Molecules. Academic Press, New York (1992)
[20] Martínez-Finkelshtein, A., Sánchez-Lara, J.F.: Shannon entropy of symmetric Pollaczek polynomials. J. Approx. Theory 145(1), 55–80 (2007) · Zbl 1111.33003 · doi:10.1016/j.jat.2006.06.007
[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
[22] Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, New York (1993) · Zbl 0891.94008
[23] Sidi, A.: Euler-Maclaurin expansions for integrals with endpoint singularities: a new perspective. Numer. Math. 98(2), 371–387 (2004) · Zbl 1081.65003 · doi:10.1007/s00211-004-0539-4
[24] Sondow, J., Weisstein, E.W.: Riemann zeta function. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannZetaFunction.html
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.