Optimization of the Euler-Maclaurin type quadrature formula in the Hilbert space of periodic functions

Abstract

In this paper, we address the construction of an optimal quadrature formula in the sense of Sard within a Hilbert space composed of periodic, complex-valued functions, specifically for the purpose of numerically approximating Fourier integrals. The quadrature formula is expressed as
a linear combination of the function’s values taken at nodes of a uniform grid. An upper estimate of the quadrature error is obtained through the norm of the associated error functional, which is evaluated using the CauchySchwarz inequality. To determine this norm explicitly, the method relies on the concept of an extremal function. The extremal function corresponding to the error functional is identified by applying the Riesz representation theorem, which guarantees that the norm of this function coincides with the norm of the error functional in the dual (conjugate) space. Ultimately, the resulting expression for the norm of the error functional takes the form of a multivariate quadratic function dependent on the coefficients of the quadrature formula. In addition, this work determines the optimal coefficients of a derived quadrature formula. Using these optimal coefficients, the exponentialweighted integrals of several functions have been approximately evaluated.