Quadrature Formulas of Gaussian Type for Fast Summation of Trigonometric Series

Yıl 2019, Cilt: 2 Sayı: 4, 168 - 182, 01.12.2019

Gradimir Mılovanovıc

https://doi.org/10.33205/cma.613948

Cited By: 1

Öz

A summation/integration method for fast summing trigonometric series is presented. The basic idea in this method is to transform the series to an integral with respect to some weight function on $\RR_+$ and then to approximate such an integral by the appropriate quadrature formulas of Gaussian type. The construction of these quadrature rules, as well as  the corresponding orthogonal polynomials on $\RR_+$, are also considered. Finally, in order to illustrate the efficiency of the presented  summation/integration method two numerical examples are included.

Anahtar Kelimeler

Summation, trigonometric series, Gaussian quadrature rule, orthogonal polynomial, three-term recurrence relation, weight function, Laplace transformation

Destekleyen Kurum

Serbian Academy of Sciences and Arts, Belgrade, Serbia

Proje Numarası

Φ−96.

Teşekkür

The author was supported in part by the Serbian Academy of Sciences and Arts, Project No. Φ−96.

Kaynakça

• T. S. Chihara: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.
• L. Carlitz: Bernoulli and Euler numbers and orthogonal polynomials. Duke Math. J. 26 (1959), 1-16.
• A. S. Cvetkovic and G. V. Milovanovic: The Mathematica package ``OrthogonalPolynomials''. Facta Univ. Ser. Math. Inform. 19 (2004), 17-36.
• W. Gautschi: On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), 289-317.
• W. Gautschi: A class of slowly convergent series and their summation by Gaussian quadrature. Math. Comp. 57 (1991), 309-324.
• W. Gautschi: Orthogonal polynomials: applications and computation, Acta Numerica (1996), 45-119.
• W. Gautschi: Orthogonal Polynomials: Computation and Approximation, Clarendon Press, Oxford, 2004.
• W. Gautschi: The spiral of Theodorus, numerical analysis, and special functions. J. Comput. Appl. Math. 235 (2010), 1042-1052.
• W. Gautschi and G. V. Milovanovic: Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series. Math. Comp. 44 (1985), 177-190.
• G. H. Golub and J. H. Welsch: Calculation of Gauss quadrature rule, Math. Comp. 23 (1969), 221-230.
• G. Mastroianni and G. V. Milovanovic: Interpolation Processes - Basic Theory and Applications. Springer Monographs in Mathematics, Springer - Verlag, Berlin - Heidelberg, 2008.
• G. V. Milovanovic: Summation of series and Gaussian quadratures, In: Approximation and Computation (R.V.M. Zahar, ed.), pp. 459-475, ISNM Vol. 119, Birkhauser Verlag, Basel-Boston-Berlin, 1994.
• G. V. Milovanovic: Summation of series and Gaussian quadratures, II. Numer. Algorithms 10 (1995), 127-136.
• G. V. Milovanovic: Summation processes and Gaussian quadratures. Sarajevo J. Math.7 (20) (2011), 185-200.
• G. V. Milovanovic: Quadrature processes and new applications. Bull. Cl. Sci. Math. Nat. Sci. Math. 38 (2013), 83-120.
• G. V. Milovanovic: Methods for computation of slowly convergent series and finite sums based on Gauss-Christoffel quadratures, Jaen J. Approx. 6 (2014), 37-68.
• G. V. Milovanovic: On summation/integration methods for slowly convergent series. Stud. Univ. Babes-Bolyai Math. 61 (2016), 359-375.
• G. V. Milovanovic: Chapter 11: Orthogonal polynomials on the real line, In: Walter Gautschi: Selected Works with Commentaries, Volume 2 (C. Brezinski, A. Sameh, eds.), pp. 3--16, Birkhauser, Basel, 2014.
• G. V. Milovanovic: Chapter 23: Computer algorithms and software packages, In: Walter Gautschi: Selected Works with Commentaries, Volume 3 (C. Brezinski, A. Sameh, eds.), pp. 9-10, Birkhauser, Basel, 2014.
• G. V. Milovanovic: Construction and applications of Gaussian quadratures with nonclassical and exotic weight functions. Stud. Univ. Babes-Bolyai Math. 60 (2015), 211-233.
• G. V. Milovanovic and A. S. Cvetkovic: Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type. Math. Balkanica 26 (2012), 169-184.
• A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev: Integrals and Series. Vol. 1. Elementary functions. Gordon & Breach Science Publishers, New York, 1986.
• T. J. Stieltjes: Sur quelques integrals definies et leur developpement en fractions continues. Quart. J. Math. 24 (1890), 370-382 [Oeuvres. Vol. 2. Noordhoff, Groningen, 1918, pp. 378-394].
• J. Touchard: Nombres exponentiels et nombres de Bernoulli. Canad. J. Math. 8 (1956), 305-320.