On an Optimal Quadrature Formula in a Hilbert Space of Periodic Functions

Abstract

The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals ∫02πeiωxφ(x)dx in the Sobolev space H˜2m. Here, H˜2m is the Hilbert space of periodic and complex-valued functions whose m-th generalized derivatives are square-integrable. Here, firstly, in order to obtain an upper bound for the error of the quadrature formula, the norm of the error functional is calculated. For this, the extremal function of the considered quadrature formula is used. By minimizing the norm of the error functional with respect to the coefficients, an optimal quadrature formula is then obtained. Using the explicit form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula is calculated. The convergence of the constructed optimal quadrature formula is investigated, and it is shown that the rate of convergence of the optimal quadrature formula is O(hm) for |ω|<N and O(|ω|−m) for |ω|≥N. Finally, we present numerical results of comparison for absolute errors of the optimal quadrature formula with the exp(iωx) weight in the case m=2 and the Midpoint formula. There, one can see the advantage of the optimal quadrature formulas.