On an optimal quadrature formula in Sobolev space $L_2^{(m)} (0,1)$
This provides a theoretical optimality result for a specific class of quadrature formulas, but is incremental as it extends known techniques to a particular functional setting.
The paper constructs optimal quadrature formulas in the Sobolev space L2^(m)(0,1) using function values at nodes and first derivative values at endpoints, deriving optimal coefficients and error norm for any N and m≥2. For m=2 and m=3, the Euler-Maclaurin formula is shown to be optimal.
In this paper in the space $L_2^{(m)}(0,1)$ the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculated for arbitrary fixed $N$ and for any $m\geq 2$. It is shown that when $m=2$ and $m=3$ the Euler-Maclaurin quadrature formula is optimal.