Optimal quadrature formulas for oscillatory integrals in the Sobolev space

Abstract

This work studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space L_{2}^{(m)}(0,1) for numerical calculation of Fourier coefficients. Using Sobolev’s method, we obtain new sine and cosine weighted optimal quadrature formulas of such type for N + 1geq m, where N + 1 is the number of nodes. Then, explicit formulas for the optimal coefficients of optimal quadrature formulas are obtained. The obtained optimal quadrature formulas in L_{2}^{(m)}(0,1) space are exact for algebraic polynomials of degree (m-1).