Knotenwerte

Knotenwerte are the numerical values of a function at designated points, called Knoten or nodes, within a given interval or domain. In interpolation theory, the Knotenwerte together with the Knoten form the data set that defines an interpolating polynomial or spline. Each Knotenwert corresponds uniquely to its Knoten, providing the ordinate that the interpolation function must reproduce exactly at that point.

Historically, the concept of Knotenwerte dates back to the early 19th‑century work on polynomial interpolation by Cauchy and later by Lagrange. Their formal use was refined in the development of spline theory in the 20th century, where Knotenwerte play a key role in ensuring smoothness across piecewise polynomial segments. In the finite element method, Knotenwerte represent nodal displacements or stresses in structural analysis; solving the governing equations yields the unknown Knotenwerte that satisfy the boundary conditions and loads.

In practice, Knotenwerte are usually measured or computed data points, such as temperature readings along a rod, observed values from a physical experiment, or sample points in a discrete dataset. The interpolation function constructed from them can be evaluated at any point within the domain, providing estimates or predictions.

The accuracy of the resulting interpolant depends on the distribution of Knoten and the magnitude of their values. Erratic or noisy Knotenwerte may lead to overfitting or oscillatory artifacts, illustrating the importance of careful data selection. For robust applications, techniques such as weighted least squares or regularized splines incorporate Knotenwerte while mitigating such issues.

Overall, Knotenwerte are fundamental to numerical approximation and simulation, encapsulating the essential information from which continuous models are reconstructed.