Optimal Quadrature Formulas with Positive Coefficients in $L_2^{(m)}(0,1)$ Space
This work provides a theoretical improvement for numerical integration in Sobolev spaces by ensuring positivity of coefficients, which is important for applications requiring non-negative weights.
The authors derive explicit optimal quadrature formulas with positive coefficients in the Sobolev space L2^{(m)}(0,1) for a specific node distribution, and compute the norm of the error functional. They compare these formulas with known optimal ones, showing that the new formulas have positive coefficients.
In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the nodes (1.5) are investigated. For optimal coefficients explicit form are obtained and norm of the error functional is calculated. In particular, by choosing parameter $η_0$ in (1.5) the optimal quadrature formulas with positive coefficients are obtained and compared with well known optimal formulas.