Polynomial interpolation and node geometry

For n+1 distinct real nodes, there is exactly one polynomial of degree at most n taking prescribed values at those nodes. The Lagrange basis builds it from polynomials that equal one at their own node and zero at every other node. Existence follows by adding the weighted basis functions. Uniqueness follows because the difference of two candidates would have n+1 distinct zeros despite degree at most n. Repeated nodes require derivative conditions and a different formulation.

Writing the interpolant in monomial coefficients is mathematically valid, but solving the associated Vandermonde system can be poorly conditioned. A barycentric evaluation formula reuses node-dependent weights and usually offers a better computational route. At an exact node, return that node's supplied value rather than forming a division by zero. Stable evaluation cannot rescue information lost through poorly chosen nodes or noisy data; it addresses the arithmetic of evaluating a particular interpolant, not every aspect of the approximation problem.

When the target function has n+1 continuous derivatives over an interval containing the nodes and the evaluation point, the interpolation remainder contains two factors: a high derivative and the product of distances from all nodes. Neither factor is automatically small as degree increases. The famous oscillations of some equally spaced high-degree interpolants show that smoothness alone does not guarantee useful uniform convergence along that particular sequence of nodes. Exact sample matching is weaker than control between samples.

Chebyshev-type nodes cluster toward interval endpoints and control the nodal product more effectively than equally spaced global nodes. Another strategy is to keep degree modest and refine with piecewise polynomials or splines. Choice depends on smoothness, data noise, desired derivatives, and whether new evaluations are available. Extrapolation outside the sampled interval is a separate and often much more sensitive task. An interior accuracy observation should never be silently promoted into an exterior prediction guarantee.

Because the interpolant depends linearly on its sample values, a perturbation in each value passes through the same basis. The sum of the absolute basis values bounds worst-case amplification at a point. Its maximum over the interval is the Lebesgue constant. This separates data sensitivity from the target function's approximation error. Even an exact evaluator cannot avoid the amplification permitted by the chosen interpolation operator. Node geometry therefore affects both approximation quality and robustness to measurement noise.

In practice, inspect errors at independent test points rather than just at the fitting nodes. Compare refinements that change nodes or local mesh spacing, and test any requested derivatives separately because differentiation can amplify interpolation errors. If observations are noisy, least squares or regularized fitting may reflect the measurement model better than forcing exact passage through every point. The correct goal is a controlled approximation to the quantity of interest, not the largest possible degree or a visually perfect fit to training samples.