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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. (English) Zbl 1405.33012

The Legendre polynomials satisfy the Bernstein’s inequality \((1-x^2)^{1/4}|P_n(x)|\leq 2/\sqrt{\pi(2n+1)}\), for any \(x\in[-1,1]\). The authors are mainly interested in obtaining uniform estimates for certain expressions of the form \((-x)^a(1+x)^b|P_n^{(\alpha,\beta)}(x)|\) on \([-1,1]\), where \(P_n^{(\alpha,\beta)}\) are the Jacobi polynomials. The obtained estimates allowed them to investigate the dispersive estimates of certain Schrödinger equations whose Hamiltonian is a discrete Laguerre operator \(H_\alpha\) acting in \(\ell^2(\{0,1,2,\ldots\})\). This is based on the fact that the kernel of the evolution group \(\text{e}^{\text{i}tH_\alpha}\) can be expressed by means of the Jacobi polynomials.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
81U30 Dispersion theory, dispersion relations arising in quantum theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Software:

DLMF; BIJ
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References:

[1] Acatrinei, C., Noncommutative radial waves, J. Phys. A: Math. Theor., 41, (2008) · Zbl 1139.81388
[2] Acatrinei, C., Discrete nonlocal waves, J. High Energy Phys., 02, (2013) · Zbl 1342.81629
[3] Akhiezer, N. I., The classical moment problem and some related questions in analysis, (1965), Oliver and Boyd Ltd. Edinburgh, London · Zbl 0135.33803
[4] Antonov, V. A.; Holševnikov, K. V., An estimate of the remainder in the expansion of the generating function for the Legendre polynomials (generalization and improvement of Bernstein’s inequality), Vestnik Leningrad Univ. Math., 13, 163-166, (1981) · Zbl 0466.33007
[5] Basu, D.; Wolf, K. B., The unitary irreducible representations of \(\operatorname{SL}(2, \mathbb{R})\) in all subgroup reductions, J. Math. Phys., 23, 189-205, (1982) · Zbl 0512.22015
[6] Burq, N.; Dyatlov, S.; Ward, R.; Zworski, M., Weighted eigenfunction estimates with applications to compressed sensing, SIAM J. Math. Anal., 44, 5, 3481-3501, (2012) · Zbl 1262.41004
[7] Chen, T.; Fröhlich, J.; Walcher, J., The decay of unstable noncommutative solitons, Comm. Math. Phys., 237, 243-269, (2003) · Zbl 1037.81094
[8] Chow, Y.; Gatteschi, L.; Wong, R., A Bernstein-type inequality for the Jacobi polynomial, Proc. Amer. Math. Soc., 121, 703-709, (1994) · Zbl 0802.33010
[9] Dunkl, C. F., The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc., 179, 331-348, (1973) · Zbl 0241.43003
[10] Dunkl, C. F.; Xu, Y., Orthogonal polynomials of several variables, Encyclopedia Math. Appl., vol. 155, (2014), Cambridge Univ. Press · Zbl 1317.33001
[11] Erdélyi, A., Higher transcendental functions, vol. 1, (1953), McGraw-Hill New York
[12] Erdélyi, A., Tables of integral transforms, vol. 1, (1954), McGraw-Hill New York
[13] Erdélyi, T.; Magnus, A. P.; Nevai, P., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal., 25, 602-614, (1994) · Zbl 0805.33013
[14] Förster, K.-J., Inequalities for ultraspherical polynomials and application for quadratures, J. Comput. Appl. Math., 49, 59-70, (1993) · Zbl 0792.33006
[15] Gautschi, W., How sharp is Bernstein’s inequality for Jacobi polynomials?, Electron. Trans. Numer. Anal., 36, 1-8, (2009/2010) · Zbl 1189.33016
[16] Gopakumar, R.; Minwalla, S.; Strominger, A., Noncommutative solitons, J. High Energy Phys., 05, (2000)
[17] Haagerup, U.; Kraus, J., Approximation properties for group \(C^\ast\)-algebras and group von Neumann algebras, Trans. Amer. Math. Soc., 344, 667-699, (1994) · Zbl 0806.43002
[18] Haagerup, U.; de Laat, T., Simple Lie groups without the approximation property, Duke Math. J., 162, 925-964, (2013) · Zbl 1266.22008
[19] Haagerup, U.; de Laat, T., Simple Lie groups without the approximation property II, Trans. Amer. Math. Soc., 368, 3777-3809, (2016) · Zbl 1345.22004
[20] Haagerup, U.; Schlichtkrull, H., Inequalities for Jacobi polynomials, Ramanujan J., 33, 227-246, (2014) · Zbl 1306.33018
[21] Koornwinder, T. H., The addition formula for Jacobi polynomials II. the Laplace type integral and the product formula, (1972), Mathematisch Centrum Amsterdam, 29 pp. · Zbl 0247.33018
[22] Koornwinder, T. H., The addition formula for Jacobi polynomials III. completion of the proof, (1972), Mathematisch Centrum Amsterdam, 11 pp. · Zbl 0247.33019
[23] Koornwinder, T. H., Two-variable analogues of the classical orthogonal polynomials, (Askey, R. A., Theory and Applications of Special Functions, (1975), Acad. Press New York, San Francisco, London), 434-495
[24] Koornwinder, T. H., Group theoretic interpretations of Askey’s scheme of hypergeometric orthogonal polynomials, (Alfaro, M.; etal., Orthogonal Polynomials and Their Applications, Lecture Notes in Math., vol. 1329, (1988), Springer-Verlag Berlin), 46-72 · Zbl 0654.33006
[25] Koornwinder, T. H., Representations of \(\operatorname{SU}(2)\) and Jacobi polynomials
[26] Koornwinder, T. H.; Schwartz, A. L., Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle, Constr. Approx., 11, 537-567, (1997) · Zbl 0937.33009
[27] Kostenko, A.; Teschl, G., Dispersion estimates for the discrete Laguerre operator, Lett. Math. Phys., 106, 4, 545-555, (2016) · Zbl 1334.35265
[28] Kostenko, A.; Teschl, G.; Toloza, J. H., Dispersion estimates for spherical Schrödinger equations, Ann. Henri Poincaré, 17, 11, 3147-3176, (2016) · Zbl 1360.34175
[29] Kovařík, H.; Truc, F., Schrödinger operators on a half-line with inverse square potentials, Math. Model. Nat. Phenom., 9, 5, 170-176, (2014) · Zbl 1320.47047
[30] Krasikov, I., An upper bound on Jacobi polynomials, J. Approx. Theory, 149, 116-130, (2007) · Zbl 1173.33007
[31] Krasikov, I., On erdélyi-Magnus-nevai conjecture for Jacobi polynomials, Constr. Approx., 28, 113-125, (2008) · Zbl 1173.33008
[32] Krueger, A. J.; Soffer, A., Structure of noncommutative solitons: existence and spectral theory, Lett. Math. Phys., 105, 1377-1398, (2015) · Zbl 1327.35328
[33] Krueger, A. J.; Soffer, A., Dynamics of noncommutative solitons I: spectral theory and dispersive estimates, Ann. Henri Poincaré, 17, 1181-1208, (2016) · Zbl 1345.81067
[34] Krueger, A. J.; Soffer, A., Dynamics of noncommutative solitons II: spectral theory, dispersive estimates and stability · Zbl 1345.81067
[35] Lafforgue, V.; de la Salle, M., Non commutative \(L^p\) spaces without the completely bounded approximation property, Duke Math. J., 160, 71-116, (2011) · Zbl 1267.46072
[36] Lohöfer, G., Inequalities for Legendre functions and Gegenbauer functions, J. Approx. Theory, 64, 226-234, (1991) · Zbl 0731.41020
[37] Lorch, L., Alternative proof of a sharpened form of Bernstein’s inequality for Legendre polynomials, Appl. Anal., 14, 3, 237-240, (1982/1983) · Zbl 0505.33007
[38] Lorch, L., Inequalities for ultraspherical polynomials and the gamma function, J. Approx. Theory, 40, 2, 115-120, (1984) · Zbl 0532.33007
[39] Olver, F. W.J., NIST handbook of mathematical functions, (2010), Cambridge University Press Cambridge · Zbl 1198.00002
[40] Rauhut, H.; Ward, R., Sparse recovery for spherical harmonic expansions, (Proceedings of the 9th International Conference on Sampling Theory and Applications, SampTA 2011, Singapore, (2011))
[41] Sally, P. J., Analytic continuation of the irreducible unitary representations of the universal covering group of \(\operatorname{SL}(2, \mathbb{R})\), Mem. Amer. Math. Soc., vol. 69, (1967), Amer. Math. Soc. · Zbl 0157.20702
[42] Szegö, G., Orthogonal polynomials, (1975), Amer. Math. Soc. Providence, RI · JFM 65.0278.03
[43] Teschl, G., Jacobi operators and completely integrable nonlinear lattices, Math. Surveys Monogr., vol. 72, (2000), Amer. Math. Soc. Rhode Island · Zbl 1056.39029
[44] Teschl, G., Mathematical methods in quantum mechanics; with applications to Schrödinger operators, (2014), Amer. Math. Soc. Rhode Island · Zbl 1342.81003
[45] Vilenkin, N. Ja., Special functions and the theory of group representations, (1968), Amer. Math. Soc. Providence
[46] Vilenkin, N. Ja.; Klimyk, A. U., Representation of Lie groups and special functions, vol. 1, (1991), Kluwer Dordrecht · Zbl 0742.22001
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