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In the present paper the optimal quadrature formula for approximate evaluation of Fourier coefficients 
is constructed for functions of the space W2(1,0)[0,1] . At the same time the explicit formulas for optimal coefficients,
which are very useful in applications, are obtained. The obtained formula is exact for the function. In
particular, as consequences of the main result the new optimal quadrature formulas for approximate evaluation of
integrals     are obtained.

  • Web Address
  • DOI
  • Date of creation in the UzSCI system13-11-2019
  • Read count500
  • Date of publication04-11-2015
  • Main LanguageIngliz
  • Pages71-77
English

In the present paper the optimal quadrature formula for approximate evaluation of Fourier coefficients 
is constructed for functions of the space W2(1,0)[0,1] . At the same time the explicit formulas for optimal coefficients,
which are very useful in applications, are obtained. The obtained formula is exact for the function. In
particular, as consequences of the main result the new optimal quadrature formulas for approximate evaluation of
integrals     are obtained.

Ўзбек

Ushbu ishda  fazoda
Furе koeffitsiyеntlarini taqribiy hisoblash uchun optimal kvadratur
formula qurilgan. Bunda optimal koeffitsiyеntlar uchun oshkor formulalar olingan bo’lib, bu formulalar qo’llash uchun
juda qulaydir. Olingan optimal kvadratur formula funksiya uchun aniqdir. Xususan, asosiy natijadan
 intеgrallarni taqribiy hisoblash uchun yangi optimal kvadratur formulalar
olindi.
 

Ўзбек

В настоящей работе построена оптимальная квадратурная формула для приближенного вычисления
коэффициентов Фурье
 функций из пространства  . При этом получены явные
формулы для оптимальных коэффициентов, которые очень удобны для применения. Полученная формула
точна для . В частности, как следствия из основного результата, получены новые оптимальные
квадратурные формулы для приближенного вычисления интегралов .
 

№ Author name position Name of organisation
1 Boltayev N.D. professor O'zbekiston Milliy universitet
2 Hayotov A.R. Fizika-matematika fanlari doktori O'zbekiston Milliy universitet
3 Xudoyberdiyev M.. ilmiy xodim Toshkent axborot texnologiyalari universiteti
№ Name of reference
1 Milovanovic G.V., Stanic M.P. Numerical integration of highly oscillating functions // Analytic Number Theory, Approximation Theory, and Special Functions. – 2014. – Pp. 613-649.
2 Filon L.N.G. On a quadrature formula trigonometric integrals // Proc. Roy. Soc. – Edinburgh. – 1928. – Pp.38-47.
3 Sard A. Best approximate integration formulas, best approximate formulas // Amer. J. Math. LXXI. – 1949. – Pp. 80-91.
4 Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas in the sense of Sard in W2( , 1) m m (0,1) space // Calcolo. – 2014. – № 51. – Pp. 211-243.
5 Sobolev S.L. Introduction to the Theory of Cubature formulas. – Moscow: Nauka, 1974 – 808 p.
6 Sobolev S.L. The coefficients of optimal quadrature formulas // Selected works of S.L. Sobolev. – Berlin: Springer, 2006). – Pp. 561-566.
7 Shadimetov Kh.M., Hayotov A.R. Construction of the discrete analogue of the differential operator 2 2 2 2 2 2 m m m m d d dx dx   // Uzbek Mathematical Journal. – Tashkent, 2004. – № 2. – Pp. 85-95.
8 Shadimetov Kh.M., Hayotov A.R. Properties of the discrete analogue of the differential operator 2 2 2 2 2 2 m m m m d d dx dx   // Uzbek Mathematical Journal. – Tashkent, 2004. – № 4. – Pp. 72-83.
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