Chebyshev nodes and interpolation stability

For degree-at-most-n interpolation on $[-1,1]$, use n+1 distinct nodes. The roots of $T_{n+1}$ are cosines of equally spaced half-offset angles; none is an endpoint. Another common grid uses extrema of $T_n$, including both endpoints when $n\ge1$. The names first-kind and second-kind points sometimes track these conventions, so writing the formula is safer than relying only on a label. Both families cluster near the interval ends rather than remaining equally spaced in x.

The identity $T_k(\cos\theta)=\cos(k\theta)$ converts a regular angular grid into a nonuniform spatial grid. Close to zero angle, cosine changes quadratically, which explains the endpoint clustering. An affine transformation carries either family to a different finite interval. This is a design decision about where information is collected; changing the nodes after measurements have already been made is generally not free and may require a different approximation method for the available data.

For $f\in C^{n+1}([-1,1])$, the degree-at-most-n interpolant has the familiar derivative remainder: the error at x is a derivative value at an intermediate point times the monic product of x minus every node, divided by a factorial. The intermediate point may depend on x. For the root grid, this monic product is $2^{-n}T_{n+1}(x)$, since that Chebyshev polynomial has the correct roots and leading coefficient $2^n$. Its maximum magnitude is therefore exactly $2^{-n}$.

Among monic polynomials of degree n+1, this scaled Chebyshev polynomial has the smallest uniform norm on the interval. To see the obstruction to improvement, subtract a hypothetical smaller monic polynomial: the leading terms cancel, leaving degree at most n, but the n+2 alternating extrema force at least n+1 roots of the difference. This proves an optimal nodal-product statement. It does not say the resulting interpolant minimizes the actual error for every target, because the derivative factor also depends on the function and evaluation point.

Approximation error asks how the exact interpolant differs from the exact function. Conditioning asks how perturbing the nodal values changes that interpolant, measured later by the Lebesgue constant. Algorithmic stability asks whether floating-point evaluation behaves like a nearby well-posed data problem. A good node set improves the first two in many important cases, while a suitable formula is still needed for the third. Solving a poorly scaled monomial Vandermonde system can waste the benefit of carefully chosen nodes.

For Chebyshev root and extrema grids, Lebesgue constants grow only logarithmically with degree, unlike the severe growth of equally spaced polynomial grids. This supports near-best approximation bounds but is not a universal promise of exponential convergence. Analytic continuation into a suitable complex neighborhood can produce geometric rates; finite smoothness generally produces algebraic rates. A discontinuous target cannot converge uniformly to continuous polynomials on an interval containing its jump, irrespective of how intelligently the nodes are placed.