An optimal interpolation formula with derivatives in Sobolev space
Abstract
This paper discusses the construction of an optimal interpolation formula intended for approximating functions in the Hilbert space $L_2^{(3)}(0,1)$. This space covers square-integrable functions with the third generalized derivative in the interval $[0,1]$. The interpolation formula is a linear combination of the function values, their first and second derivatives at equally spaced nodes in the interval $[0,1]$. The coefficients are determined by minimizing the norm of the error functional in the conjugate space $ L_2^{(3)*}(0,1)$. This error functional is defined as the discrepancy between the function and its approximation. The key results of the study include explicit expressions for the coefficients and the norm of the error functional. The optimization problem was methodically formulated and solved, resulting in a system of linear equations for the coefficients. Analytical solutions were obtained that give a clear expression for optimal coefficients. In addition, integrating the obtained optimal interpolation formula over the interval $[0,1]$ leads to the Euler-Maclaurin quadrature formula. The application of these results in estimating the error of the interpolation formula for functions from $L_2^{(3)}(0 ,1)$ is demonstrated.