An optimal quadrature formula for approximating Fourier integrals in a Hilbert space

This paper presents the construction process of the optimal quadrature formula for the approximate calculation of the integrals ∫abe2πiωxφ(x)dx with real ω in the Hilbert space of complex-valued periodic functions. Here, initially, in order to obtain a sharp upper bound for the error of the quadrature formula, the norm of the error functional is calculated. For this, the extremal function of the error functional for the quadrature formula is used. Then, by minimizing the norm of the error functional with respect to the coefficients, an optimal quadrature formula is obtained. Using the explicit form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula is calculated. Finally, using this optimal quadrature formula the approximation formula for Fourier integrals ∫abe2πiωxφ(x)dx with ω∈ℝ is obtained in the interval [a, b].

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