 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 FunctionsKholmat Shadimetov (),
Abdullo Hayotov () and
Umedjon Khayriev
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 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-031-85743-0_16 Ordering information: This item can be ordered from DOI: 10.1007/978-3-031-85743-0_16 Access Statistics for this chapter More chapters in Springer Optimization and Its Applications from Springer |
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