American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: http://www.sciepub.com/journal/ajna Editor-in-chief: Emanuele Galligani
Open Access
Journal Browser
Go
American Journal of Numerical Analysis. 2016, 4(1), 1-7
DOI: 10.12691/ajna-4-1-1
Open AccessArticle

Construction of Optimal Quadrature Formula for Numerical Calculation of Fourier Coefficients in Sobolev Space L2(1)

Nurali D. Boltaev1, Abdullo R. Hayotov1, and Kholmat M. Shadimetov1

1Department of Comptational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

Pub. Date: January 27, 2016

Cite this paper:
Nurali D. Boltaev, Abdullo R. Hayotov and Kholmat M. Shadimetov. Construction of Optimal Quadrature Formula for Numerical Calculation of Fourier Coefficients in Sobolev Space L2(1). American Journal of Numerical Analysis. 2016; 4(1):1-7. doi: 10.12691/ajna-4-1-1

Abstract

In the present paper the optimal quadrature formula for approximate evaluation of Fourier coefficients is constructed for functions of the space . At the same time the explicit formulas for optimal coefficients, which are very useful in applications, are obtained. The obtained formula is exact for constant. In particular, as consequences of the main result the new optimal quadrature formulas for approximate evaluation of integrals and are obtained. Furthermore, the order of convergence of the constructed optimal quadrature formula is studied.

Keywords:
optimal quadrature formula extremal function error functional hilbert space

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Babuska, I., Vitasek, E., Prager, M. Numerical processes in differential equations. Wiley, New York, 1966.
 
[2]  Bakhvalov, N.S., Vasil'eva, L.G. Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures (Russian) USSR Computational Mathematics and Mathematical Physics. 8, 1968, 241-249.
 
[3]  Filon, L.N.G. On a quadrature formula trigonometric integrals // Proc. Roy. Soc. Edinburgh. 1928. Pp.38-47.
 
[4]  Flinn, E.A. A modification of Filon’s method of numerical integration, J. Assoc. Comp. Mach. 7 (1960) 181-184.
 
[5]  Havie, T. Remarks on an expansion for integrals of rapidly oscillation functions, BIT, 13 (1973) 16-29.
 
[6]  Iserles, A., Nørsett, S.P. On quadrature methods for highly oscillatory integrals and their implementation, BIT Numer Math 44: 755-772, 2004.
 
[7]  Iserles, A., Nørsett, S.P. Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. A (2005) 461, 1383-1399.
 
[8]  Levin, D. Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math., 67 (1996) 95-101.
 
[9]  Melenk, J.M. On the convergence of Filon quadrature, Jour of Comp and Appl Math 234 (2010) 1692-1701.
 
[10]  Milovanovic, G.V. Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures, Computers Math. Applic. Vol. 36, no. 8, (1998). 19-39.
 
[11]  Milovanovic, G.V., Stanic, M.P. Numerical integration of highly oscillating functions // Analytic Number Theory, Approximation Theory, and Special Functions. Springer, 2014. Pp. 613-649.
 
[12]  Novak, E., Ullrich, M., Wozniakowski, H. Complexity of oscillatory integration for univariate Sobolev space, Journal of Complexity, 31 (2015) 15-41.
 
[13]  Olver, S. Numerical approximation of highly oscillatory integrals, PhD dissertation, University of Cambridge, 2008.
 
[14]  Olver, Sh. Fast, numerically stable computation of oscillatory integrals with stationary points, BIT Numer Math (2010) 50: 149-171.
 
[15]  Sard, A. Best approximate integration formulas; best approximation formulas, Am. J. Math. 71 (1949) 80-91.
 
[16]  Shampine, L.F. Efficient Filon method for oscillatory integrals, Appl Math and Comp 221 (2013) 691-702.
 
[17]  Shadimetov, Kh.M. The discrete analogue of the differential operator and its construction, Quest. Comput. Appl. Math., Tashkent, 1985, pp. 22-35 (2010).
 
[18]  Shadimetov, Kh.M. Weight optimal cubature formulas in Sobolev's periodic space, (Russian) Siberian J. Numer. Math. -Novosibirsk, v.2, no. 2 (1999) 185-196.
 
[19]  Sobolev, S.L. Introduction to the Theory of Cubature formulas. – Moscow: Nauka, (Russian) 1974-808 p.
 
[20]  Sobolev, S.L. The coefficients of optimal quadrature formulas // Selected works of S.L. Sobolev. – Berlin: Springer, 2006).Pp. 561-566.
 
[21]  Xu, Z., Milovanovic, G.V., Xiang, S. Efficient computation of highly oscillatory integrals with Henkel kernel, Appl. Math. and Comp. 261 (2015) 312-322.