Optimal Approximation of Fourier Integrals by the Phi-Function Method
Keywords:
Hilbert space, phi-function method, the optimal quadrature formula, error of the quadrature formulaAbstract
This paper investigates the construction of an optimal quadrature formula for Fourier integrals of ????(????) function using the method of ????-functions. The function ????(????) belongs to the Hilbert space, which consists of absolutely continuous functions with quadratically integrable first-order derivatives. We explore the problem of optimality in the sense of Sard, aiming to minimize the supremum of the absolute error |????????(????)| over all functions in. The method of ????-functions provides a means to derive the coefficients ???????? that achieve this minimal error. Our work details the ????-function method and its application to constructing optimal quadrature formulas, presenting a systematic approach to minimizing the quadrature error.
References
N.S. Bakhvalov, L.G. Vasil’eva 1968. Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures, Zh. Vychisl. Mat. Mat. Fiz., – 8(1), – P. 175–181. (Russian).
A. Iserles, S.P. Nшrsett 2005. Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. A, – 461, – P. 1383–1399.
G.V. Milovanović 1998. Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures, Computers Math. Applic., – 36(8), – P. 19–39.
G.V. Milovanović, M.P. Stanić 2014. Numerical integration of highly oscillating functions, in *Analytic Number Theory, Approximation Theory and Special Functions* (Edited by G.V. Milovanović), Springer-Verlag, New York, – P. 613–649.
S. Olver 2008. Numerical Approximation of Highly Oscillatory Integrals, PhD Dissertation, University of Cambridge,
Z. Xu, G.V. Milovanović, S. Xiang 2015. Efficient computation of highly oscillatory integrals with Henkel kernel, Appl. Math. Comput., – 261, – P. 312–322.
N.D. Boltaev, A.R. Hayotov, Kh.M. Shadimetov 2016. Construction of optimal quadrature formula for numerical calculation of Fourier coefficients in Sobolev space ????(1)2 , Amer. J. Numer. Anal., – 4, – P. 1–7.
Boltaev N.D., Hayotov A.R., Milovanović G.V., Shadimetov Kh.M. 2017. Optimal quadrature formulas for numerical evaluation of Fourier coefficients in ????(????,????−1) 2 //, – vol. 7, no. 4, – P. 1233–1266.
Boltaev N.D., Hayotov A.R., Khudayberdiev M. 2015. Optimal quadrature formula for approximate calculation of Fourier coefficients in????(1,0)2 space //, – Tashkent, – vol.1, – no. 1, – P. 71–77.
T. Cˇatinaş, Gh. Coman 2006. Optimal quadrature formulas based on the ????-function method, Stud. Univ., Babeş-Bolyai Math., – 51(1), – P. 49–64.
Gh. Coman 1972. Formule de cuadrature de type Sard, Stud. Univ., Babeş-Bolyai Math.-mech., – 17(2), – P. 73–77.
Gh. Coman 1972. Monosplines and optimal quadrature formulae, Lp. Rend. Mat., – 5(6), – P. 567–577.
A. Ghizzetti, A. Ossicini 1970. Quadrature Formulae, Academie Verlag, Berlin,
A.R. Hayotov, S. Jeon, C.O. Lee 2020. On an optimal quadrature formula for approximation of Fourier integrals in the space ????(1)2 , J. Comput. Appl. Math., – 372, 112713.
A.R. Hayotov, S. Jeon, Kh.M. Shadimetov 2021. Application of optimal quadrature formulas for reconstruction of CT images, J. Comput. Appl. Math., – 388, 113313.
A.R. Hayotov, S. Jeon, Ch.-O. Lee, Kh.M. Shadimetov 2021. Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction, Filomat, – 35(12), – P. 4177–4195. – DOI: 10.2298/FIL2112177H.
A.R. Hayotov, S.S. Babaev 2021. Optimal quadrature formulas for computing of Fourier integrals in ????(????,????−1) 2 space, AIP Conference Proceedings, – 2365, 020021.
P. Kцhler 1988. On the Weights of Sard’s Quadrature Formulas, Calcolo, – 25(3), – P. 169–186.
F. Lanzara 2000. On optimal quadrature formulae, Journal of Ineq. Appl., – 5, – P. 201–225.
L.F. Meyers, A. Sard 1950. Best approximate integration formulas, J. Math and Phys., – XXIX, – P. 118–123.
A. Sard 1949. Best approximate integration formulas, best approximate formulas, American J. of Math., – LXXI, – P. 80–91.
A. Sard 1963. Linear approximation, American Math. Society, Province, Rhode Island,
I.J. Schoenberg 1965. On monosplines of least deviation and best quadrature formulae, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., – 2, – P. 144–170.
I.J. Schoenberg 1966. On monosplines of least square deviation and best quadrature formulae II, SIAM J. of Numer. Anal., – 3, – P. 321–328.
I.J. Schoenberg, S.D. Silliman 1973. On semicardinal quadrature formulae, Math. Comp., – 126, – P. 483–497.
