Construction of an optimal quadrature formula for the approximate calculation of Fourier integrals using the φ-function method

Numerical integration of definite integrals is important in fundamental and applied sciences. The error of approximate integral calculations depends on the initial data and specific requirements, which leads to the imposition of various conditions on the obtained calculations. In this paper, we consider the problem of constructing an optimal quadrature formula for the approximate calculation of Fourier integrals using the $\varphi$-function method. The error of the quadrature formula is estimated from above using the integral of the square of the function $\varphi$ from the Hilbert space. Next, a function $\varphi$ is chosen for which the integral of the square of the function on this interval takes the smallest value. The coefficients of the optimal quadrature formula are calculated using the obtained $\varphi$ function. The optimal quadrature formula is exact for the functions $e^{\sigma x}$ and $e^{-\sigma x}$, where $\sigma$ is a nonzero real parameter.