On an Optimal Interpolation Formula with Derivative in a Hilbert Space

Abstract

This study is devoted to the development of a system of equations for determining the coefficients of optimal interpolation formulas that integrate derivative information within a Hilbert space framework. Conventional interpolation methods, which rely solely on pointwise function values, often prove inadequate for functions exhibiting complex or rapidly varying behavior. To overcome this limitation, the proposed approach incorporates derivative data into the interpolation process, enhancing both the stability and accuracy of the resulting formulas. By formulating the interpolation problem within a Hilbert space, we establish a robust framework for deriving these optimal coefficients. The core analytical contribution lies in the formulation of a system of equations, derived through variational principles and leveraging tools such as the Riesz representation theorem and convolution operations. Solving this system enables the explicit computation of the optimal coefficients, which are readily applicable to practical interpolation tasks. This methodology is particularly significant in numerical analysis, especially in scenarios where positional data, directional vectors, or object velocities are available, as the inclusion of derivative information is both intuitive and critical. Furthermore, the approach is applicable to data approximation, signal processing, and computational contexts requiring function reconstruction from sampled or noisy data.