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The Optimal Quadrature Formula for the Approximate Calculation of Fourier Coefficients in the Space W 2 ˜ ( 2 , 1 ) $$ \widetilde {W_{2} }^{(2,1)}$$ of Periodic Functions

Author

Listed:
  • Kholmat Shadimetov

    (Tashkent State Transport University
    Uzbekistan Academy of Sciences)

  • Abdullo Hayotov

    (Uzbekistan Academy of Sciences
    Central Asian University
    Bukhara State University)

  • Umedjon Khayriev

    (Bukhara State University
    Uzbekistan Academy of Sciences)

Abstract

In this work, the process of constructing the optimal quadrature formula in the Hilbert space W ˜ 2 ( 2 , 1 ) ( 0 , 1 ] $$\widetilde {W}_{2}^{(2,1)} (0,1]$$ of complex-valued periodic functions for the numerical calculation of Fourier coefficients is studied. Here a quadrature sum consists of a linear combination of the given function values on a uniform mesh. The error of a quadrature formula is estimated from above by the functional norm of the error based on the Cauchy-Schwarz inequality. To calculate the norm, the concept of an extremal function is used. Also, the optimal coefficients of the quadrature formula are found. Furthermore, the sharp upper bound of the error of the constructed optimal quadrature formula is found, and it is shown that the order of convergence of the optimal quadrature formula is O 1 N + ω 2 $$O\left (\left (\frac {1}{N+\left |\omega \right |} \right )^{2} \right )$$ .

Suggested Citation

  • Kholmat Shadimetov & Abdullo Hayotov & Umedjon Khayriev, 2025. "The Optimal Quadrature Formula for the Approximate Calculation of Fourier Coefficients in the Space W 2 ˜ ( 2 , 1 ) $$ \widetilde {W_{2} }^{(2,1)}$$ of Periodic Functions," Springer Optimization and Its Applications,, Springer.
    • Handle: RePEc:spr:spochp:978-3-031-85743-0_16
    DOI: 10.1007/978-3-031-85743-0_16
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