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Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method

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Abstract

The paper studies Sard’s problem on construction of optimal quadrature formulas in the space W (m,0)2 by Sobolev’s method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W (m,0)2 is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue Dm(hβ) of the differential operator \({{{d^{2m}}} \over {d{x^{2m}}}} - 1\) is constructed. Further, Sobolev’s method of construction of optimal quadrature formulas in the space W (m,0)2 , which based on the discrete analogue Dm(hβ), is described. Next, for m = 1 and m = 3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained. Finally, at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W (3,0)2 for the cases m = 1 and m = 3 are presented.

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Acknowledgements

The work has been done while A. R. Hayotov was visiting Department of Mathematical Sciences at KAIST, Daejeon, Republic of Korea. A. R. Hayotov’s work was supported by the “Korea Foundation for Advanced Studies”/“Chey Institute for Advanced Studies” International Scholar Exchange Fellowship for academic year of 2018–2019. A. R. Hayotov is very grateful to professor Chang-Ock Lee and his research group for hospitality. The authors are very thankful to the reviewer for the helpful remarks which have improved the quality of the paper.

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Authors and Affiliations

  1. V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b, University Str., Tashkent, 100174, Uzbekistan

    Aziz Boltaev, Abdullo Hayotov & Kholmat Shadimetov

  2. Tashkent Railway Engineering Institute, 1, Odilxojaev Str., Tashkent, 100167, Uzbekistan

    Kholmat Shadimetov

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  1. Aziz Boltaev
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  2. Abdullo Hayotov
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  3. Kholmat Shadimetov
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Correspondence to Abdullo Hayotov.

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Boltaev, A., Hayotov, A. & Shadimetov, K. Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method. Acta. Math. Sin.-English Ser. 37, 1066–1088 (2021). https://doi.org/10.1007/s10114-021-9506-6

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  • DOI: https://doi.org/10.1007/s10114-021-9506-6

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