Optimal quadrature formulas for oscillatory integrals in the Sobolev space

AbstractThis 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 $N + 1\geq m$ 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 ) .