Optimal quadrature formulas for oscillatory integrals in the Sobolev space - Kholmat Shadimetov; Abdullo Hayotov; Bakhromjon Bozarov (Journal of Inequalities and Applications, 2022)
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 ) $L_{2}^{(m)}(0,1)$ 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 + 1 ≥ m $N + 1≥ m$ , where N + 1 $N + 1$ 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 ) $L_{2}^{(m)}(0,1)$ space are exact for algebraic polynomials of degree ( m − 1 ) $(m-1)$ .