The tendency for high-degree polynomial interpolants at equispaced nodes to exhibit large oscillations near the interval endpoints, producing poor uniform approximation despite convergence at many interior points.

Principle

Polynomial interpolation with equidistant nodes amplifies conditioning errors and the influence of high-degree basis polynomials near boundaries, causing nonuniform convergence.

Demonstration

Interpolating the function f(x)=1/(1+25x^2) on [-1,1] with equally spaced nodes and increasing polynomial degree yields growing oscillations at the ends, whereas using Chebyshev nodes mitigates the effect.

Misapplication

Assuming that increasing polynomial degree with uniformly spaced samples always improves uniform approximation leads to degraded results and numerical instability.

Consequence

Drives the adoption of nonuniform node distributions, piecewise interpolation (splines), or spectral methods with orthogonal polynomials to obtain stable uniform approximations.

Reversal

Replacing global high-degree polynomials on equispaced nodes by piecewise low-degree polynomials or by global expansions at Chebyshev-type nodes reverses the oscillatory behavior.

Boundary

Pertains specifically to polynomial interpolation on compact intervals with equispaced nodes; does not preclude good approximation by other bases or by polynomials with adapted node sets.

Semantic Tension

Contrasts with spectral convergence claims for orthogonal polynomial expansions where appropriate node selection yields rapid uniform convergence for smooth functions.

Synthesis

An interpolation instability phenomenon showing that naive high-degree polynomial fitting on equispaced points can worsen approximation near boundaries, motivating better node choices or local methods.