Optimal quadrature formulas for oscillatory integrals in the Sobolev space

Abstract

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 $N + 1\geq m$ , 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 ) $(m-1)$ .

Keywords

• Optimal quadrature formulas
• Fourier coefficients
• Error functional
• Optimal coefficients
• Sobolev space
• The weighted function