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

Kholmat Shadimetov (), Abdullo Hayotov () and Umedjon Khayriev
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Kholmat Shadimetov: Tashkent State Transport University
Abdullo Hayotov: Uzbekistan Academy of Sciences
Umedjon Khayriev: Bukhara State University

Chapter Chapter 16 in Analysis, Approximation, Optimization: Computation and Applications, 2025, pp 287-300 from Springer

Abstract: 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 $$O\left (\left (\frac {1}{N+\left |\omega \right |} \right )^{2} \right )$$ .

Date: 2025
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DOI: 10.1007/978-3-031-85743-0_16

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