An optimal interpolation formula of Hermite type in the Sobolev space

Abstract

This article delves into the construction of an optimal interpolation formula designed for approximating functions within the Hilbert space $L_2^{(2)}(0,1)$. This space encompasses functions that are square integrable with a second generalized derivative in the interval $[0,1]$.
The interpolation formula takes the form of a linear combination of function values and their first-order derivative at equidistant nodes within the interval $[0,1]$. The coefficients are determined by minimizing the norm of the error functional in the dual space $L_2^{(2)*}(0,1)$. This error functional is defined as the disparity between the function and its approximation.

Key outcomes of the study include explicit expressions for the coefficients and the norm of the error functional. The optimization problem is methodically formulated and solved, resulting in a system of linear equations for the coefficients. Analytical solutions are achieved, yielding a clear expression for the optimal coefficients.

Furthermore, integrating the obtained optimal interpolation formula over the interval $[0,1]$, yields the Euler-Maclaurin quadrature formula. The application of these results is demonstrated in estimating the error of the interpolation formula for functions in $L_2^{(2)}(0,1)$.