On optimal convergence rates of Laguerre polynomial expansions for piecewise functions. (English) Zbl 1514.65183
Summary: By applying Hilb type formula and generalized van der Corput type Lemmas, this paper presents optimal decay rates on the expansion coefficients for given functions expanded in the form of Laguerre polynomial series, from which it leads to the convergence rates on the spectral orthogonal projections, and derives new estimates on the errors for Gauss-Laguerre quadrature. Ample numerical experiments are carried out to validate the optimality and accuracy of these theoretical results.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 41A10 | Approximation by polynomials |
| 41A25 | Rate of convergence, degree of approximation |
Software:
ChebfunReferences:
| [1] | Golub, G. H.; Welsch, J. H., Calculation of Gauss quadrature rules, Math. Comp., 23, 221-230 (1969) · Zbl 0179.21901 |
| [2] | Glaser, A.; Liu, X.; Rokhlin, V., A fast algorithm for the calculation of the roots of special functions, SIAM J. Sci. Comput., 29, 1420-1438 (2007) · Zbl 1145.65015 |
| [3] | L.N. Trefethen, N. Hale, R.B. Platte, T.A. Driscoll, R. Pachón, Chebfun, University of Oxford, 2009, http://www.maths.ox.ac.uk/chebfun. |
| [4] | Fatyanov, A. G.; Terekhov, A. V., High-performance modeling acoustic and elastic waves using the parallel Dichotomy algorithm, J. Comput. Phys., 230, 1992-2003 (2011) · Zbl 1391.76688 |
| [5] | Terekhov, A. V., A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method, Parallel Comput., 39, 245-258 (2013) |
| [6] | Mikhailenko, B. G., Spectral Laguerre method for the approximate solution of time dependent problems, Appl. Math. Lett., 12, 105-110 (1999) · Zbl 0939.65114 |
| [7] | Mastryukov, A. F.; Mikhailenko, B. G., Numerical solution of Maxwell’s equations for anisotropic media using the Laguerre transform, Russ. Geol. Geophys., 49, 621-627 (2008) |
| [8] | Colton, D.; Wimp, J., Analytic solutions of the heat equation and some formulas for laguerre and Hermite polynomials, Complex Var. Theory Appl.: Int. J., 3, 397-412 (1984) |
| [9] | Jo, J.; Fang, Q.; Papaioannou, T.; Baker, J. D.; Dorafshar, A.; Reil, T.; Qiao, J.; Fishbein, M.; Freischlag, J.; Marcu, L., Laguerre-based method for analysis of time-resolved fluorescence data: application to in-vivo characterization and diagnosis of atherosclerotic lesions, J. Biomed. Opt., 11, Article 021004 pp. (2006) |
| [10] | Xiang, S., Asymptotics on Laguerre or Hermite polynomial expansions and their applications in Gauss quadrature, J. Math. Anal. Appl., 393, 434-444 (2012) · Zbl 1259.65058 |
| [11] | Szegő, G., Orthogonal Polynomials (1939), American Mathematical Society · JFM 65.0286.02 |
| [12] | Gil, A.; Segura, J.; Temme, N. M., Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures, Stud. Appl. Math., 140, 298-332 (2018) · Zbl 1391.65134 |
| [13] | Zaman, S.; ul Islam, S.; Khan, M. M.; Ahmad, I., New algorithms for approximation of bessel transforms with high frequency parameter, J. Comput. Appl. Math., 399, Article 113705 pp. (2022) · Zbl 1481.65045 |
| [14] | Zaman, S.; ul Islam, S.; Hussain, I., Approximation of highly oscillatory integrals containing special functions, J. Comput. Appl. Math., 365, Article 112372 pp. (2020) · Zbl 1490.65036 |
| [15] | Zaman, S.; ul Islam, S., On numerical evaluation of integrals involving oscillatory Bessel and Hankel functions, Numer. Algorithms, 82, 1325-1343 (2019) · Zbl 1477.65055 |
| [16] | ul Islam, S.; Zaman, S., Numerical methods for multivariate highly oscillatory integrals, Int. J. Comput. Math., 95, 1024-1046 (2018) · Zbl 1499.65081 |
| [17] | Xiang, S., Convergence rates on spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities, SIAM J. Numer. Anal., 59, 1374-1398 (2021) · Zbl 1465.41003 |
| [18] | Xiang, S.; Liu, G., Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities, Numer. Math., 145, 117-148 (2020) · Zbl 1511.42024 |
| [19] | Xiang, S., On van der Corput-type lemmas for Bessel and Airy transforms and applications, J. Comput. Appl. Math., 351, 179-185 (2019) · Zbl 1459.65033 |
| [20] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1965), Dover Publications |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
