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Mat. Sb., 1994, Volume 185, Number 8, Pages 3–30 (Mi msb916)  

This article is cited in 50 scientific papers (total in 51 papers)

Asymptotic behavior of the $L^p$-norms and the entropy for general orthogonal polynomials

A. I. Aptekareva, V. S. Buyarovb, I. S. Degeza

a M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University

Abstract: The limit behavior of the sequence of $L^p$-norms of orthogonal polynomials is studied. Orthogonal polynomials on both a finite interval and the entire real line are considered. As a corollary, asymptotic formulas for entropy integrals containing orthogonal polynomials are obtained.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 82:2, 373–395

Bibliographic databases:

UDC: 517.53
MSC: Primary 42C05; Secondary 42C99
Received: 20.12.1993

Citation: A. I. Aptekarev, V. S. Buyarov, I. S. Degeza, “Asymptotic behavior of the $L^p$-norms and the entropy for general orthogonal polynomials”, Mat. Sb., 185:8 (1994), 3–30; Russian Acad. Sci. Sb. Math., 82:2 (1995), 373–395

Citation in format AMSBIB
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\by A.~I.~Aptekarev, V.~S.~Buyarov, I.~S.~Degeza
\paper Asymptotic behavior of the~$L^p$-norms and the~entropy for general orthogonal polynomials
\jour Mat. Sb.
\yr 1994
\vol 185
\issue 8
\pages 3--30
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1302621}
\zmath{https://zbmath.org/?q=an:0862.46014}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 82
\issue 2
\pages 373--395
\crossref{https://doi.org/10.1070/SM1995v082n02ABEH003571}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Yanez R., Angulo J., Dehesa J., “Information Entropies of Many-Electron Systems”, Int. J. Quantum Chem., 56:5 (1995), 489–498  crossref  isi
    2. Aptekarev A., Buyarov V., WanAsche V., Deheza J., “Asymptotics of Entropy Integrals for Orthogonal Polynomials”, Dokl. Akad. Nauk, 346:4 (1996), 439–441  mathnet  mathscinet  zmath  isi
    3. Jorge Sánchez-Ruiz, “Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates”, J Math Phys (N Y ), 38:10 (1997), 5031  crossref  mathscinet  zmath  isi  elib
    4. Jorge Sánchez-Ruiz, “Asymptotic formula for the quantum entropy of position in energy eigenstates”, Physics Letters A, 226:1-2 (1997), 7  crossref  mathscinet  zmath  elib
    5. Buyarov V., “On Information Entropy for Gegenbauer Polynomials”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1997, no. 6, 8–11  mathscinet  zmath  isi
    6. Buyarov V., “On Distribution Function of Orthogonal Polynomial Values”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1998, no. 4, 3–5  mathscinet  zmath  isi
    7. Dehesa J., Yanez R., Aptekarev A., Buyarov V., “Strong Asymptotics of Laguerre Polynomials and Information Entropies of Two-Dimensional Harmonic Oscillator and One-Dimensional Coulomb Potentials”, J. Math. Phys., 39:6 (1998), 3050–3060  crossref  mathscinet  zmath  adsnasa  isi
    8. Jorge S$aacute$nchez-Ruiz, J Phys A Math Gen, 32:18 (1999), 3419  crossref  mathscinet  zmath  isi
    9. R. J. Yáñez, W. Van Assche, R. González-Férez, Jesús S. Dehesa, “Entropic integrals of hyperspherical harmonics and spatial entropy of D-dimensional central potentials”, J Math Phys (N Y ), 40:11 (1999), 5675  crossref  mathscinet  zmath  adsnasa  isi
    10. Buyarov V., Dehesa J., Martinez-Finkelshtein A., Saff E., “Asymptotics of the Information Entropy for Jacobi and Laguerre Polynomials with Varying Weights”, J. Approx. Theory, 99:1 (1999), 153–166  crossref  mathscinet  zmath  isi
    11. V S Buyarov, P López-Artés, A Martínez-Finkelshtein, W Van Assche, J Phys A Math Gen, 33:37 (2000), 6549  crossref  mathscinet  zmath  adsnasa  isi
    12. Lubinsky, DS, “Asymptotics of orthogonal polynomials: Some old, some new, some identities”, Acta Applicandae Mathematicae, 61:1–3 (2000), 207  crossref  mathscinet  zmath  isi  elib
    13. Jorge Sánchez-Ruiz, Jesús S. Dehesa, “Entropic integrals of orthogonal hypergeometric polynomials with general supports”, Journal of Computational and Applied Mathematics, 118:1-2 (2000), 311  crossref  mathscinet  zmath
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    15. He M., Ricci P., “Asymptotics for Entropy Integrals of Orthogonal Polynomials on the Unit Circle”, Nonlinear Anal.-Theory Methods Appl., 47:3 (2001), 1941–1951  crossref  mathscinet  zmath  isi
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    18. He M., Ricci P., “Information Entropy of Orthogonal Polynomials”, Appl. Math. Comput., 128:2-3 (2002), 261–274  crossref  mathscinet  zmath  isi
    19. Jorge S nchez-Ruiz, “Information entropy of Gegenbauer polynomials and Gaussian quadrature”, J Phys A Math Gen, 36:17 (2003), 4857  crossref  mathscinet  zmath  isi  elib
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    21. Eli Levin, D.S. Lubinsky, “Asymptotics for entropy integrals associated with exponential weights”, Journal of Computational and Applied Mathematics, 156:2 (2003), 265  crossref  mathscinet  zmath
    22. B. Beckermann, A. Martínez-Finkelshtein, E. A. Rakhmanov, F. Wielonsky, “Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class”, J Math Phys (N Y ), 45:11 (2004), 4239  crossref  mathscinet  zmath  adsnasa  isi
    23. Mark W. Coffey, “Asymptotic relation for the quantum entropy of momentum in energy eigenstates”, Physics Letters A, 324:5-6 (2004), 446  crossref  mathscinet  zmath  elib
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    25. M. A. Prikhod'ko, “Asymptotics of Information Entropy for the Two-Dimensional Analog of the Relativistic Hydrogen Atom in the Kozlov–Nikishin Model”, Math. Notes, 78:5 (2005), 677–692  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    27. A. Martínez-Finkelshtein, J.F. Sánchez-Lara, “Shannon entropy of symmetric Pollaczek polynomials”, Journal of Approximation Theory, 145:1 (2007), 55  crossref  mathscinet  zmath
    28. A. I. Aptekarev, J. S. Dehesa, A. Martínez-Finkelshtein, R. Yáñez, “Discrete Entropies of Orthogonal Polynomials”, Constr Approx, 2008  crossref  mathscinet  zmath  isi
    29. J. S. Dehesa, S. López-Rosa, A. Martínez-Finkelshtein, R. J. Yáñez, “Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states”, Int J Quantum Chem, 2009, NA  crossref
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    34. Bujar Xh. Fejzullahu, “Asymptotic properties and Fourier expansions of orthogonal polynomials with a non-discrete Gegenbauer–Sobolev inner product”, Journal of Approximation Theory, 162:2 (2010), 397  crossref  mathscinet  zmath
    35. J. S. Dehesa, A. Guerrero, P. Sánchez-Moreno, “Information-Theoretic-Based Spreading Measures of Orthogonal Polynomials”, Complex Anal. Oper. Theory, 2011  crossref  mathscinet
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    41. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of $L_p$-norms of Hermite polynomials”, Preprinty IPM im. M. V. Keldysha, 2012, 070, 22 pp.  mathnet
    42. S. López-Rosa, I. V. Toranzo, P. Sánchez-Moreno, J. S. Dehesa, “Entropy and complexity analysis of hydrogenic Rydberg atoms”, J. Math. Phys, 54:5 (2013), 052109  crossref  mathscinet
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    44. Fejzullahu B.X., “Mehler-Heine Formulas For Orthogonal Polynomials With Respect To the Modified Jacobi Weight”, Proc. Amer. Math. Soc., 142:6 (2014), 2035–2045  crossref  mathscinet  zmath  isi
    45. Dehesa J.S., Guerrero A., Lopez J.L., Sanchez-Moreno P., “Asymptotics (P -> Infinity) of l-P-Norms of Hypergeometric Orthogonal Polynomials”, J. Math. Chem., 52:1 (2014), 283–300  crossref  mathscinet  zmath  isi
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    47. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of $L_p$-norms of Laguerre polynomials”, Russian Math. Surveys, 70:5 (2015), 955–957  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    48. V. M. Buchstaber, V. N. Dubinin, V. A. Kaliaguine, B. S. Kashin, V. N. Sorokin, S. P. Suetin, D. N. Tulyakov, B. N. Chetverushkin, E. M. Chirka, A. A. Shkalikov, “Alexander Ivanovich Aptekarev (on his 60th birthday)”, Russian Math. Surveys, 70:5 (2015), 965–973  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    49. A. I. Aptekarev, S. A. Denisov, “Problema Steklova i otsenki ortogonalnykh mnogochlenov s vesami iz klassov $A_p(\mathbb{T})$”, Preprinty IPM im. M. V. Keldysha, 2016, 040, 19 pp.  mathnet
    50. E. D. Belega, D. N. Tulyakov, “High-dimensional oscillators and Laguerre polynomials”, Russian Math. Surveys, 72:5 (2017), 965–967  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    51. A. I. Aptekarev, “Slabye i silnye asimptotiki ortogonalnykh mnogochlenov c «peremennym» vesom”, Posvyaschaetsya 70-letiyu prezidenta RUDN V.M. Filippova, SMFN, 67, no. 3, Rossiiskii universitet druzhby narodov, M., 2021, 427–441  mathnet  crossref
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