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If interpolation at distinct sites $x_i$ in a finite-dimensional space of continuous functions is unique, its cardinal functions satisfy ${\ell }_i(x_j)={\delta }_{ij}$. The interpolation linear map is $Pf={\sum }_{i}f(x_i){\ell }_i$ and has operator norm ${\Lambda }_{\mathbf{x}}$ in the supremum norm. The upper bound is the triangle inequality. For the reverse bound choose node values to be the signs of the cardinal functions at a point maximizing their absolute sum, and extend those values by a continuous function of norm one. The polynomial reproduction error bound gives $\Vert f-Pf{\Vert }_{\infty }\le (1+{\Lambda }_{\mathbf{x}}){\inf }_{s\in \mathrm{range}P}\Vert f-s{\Vert }_{\infty }$.

For any trigonometric polynomial $t_n$ of degree at most $n$, reproduction from (a) and the operator norm bound from (b) implyTaking the infimum over $t_n$ givesThis is the polynomial reproduction error bound with $\Vert v_n\Vert \le 3$; attaining the best uniform approximation is not needed for this inequality.

Take any trigonometric polynomial $t\in {\mathcal{T}}_n$. The reproduction property in (a) givesBy (b),Taking the infimum over $t\in {\mathcal{T}}_n$ yields the polynomial reproduction error bound for this operator. With $E_n(f)={\inf }_{t\in {\mathcal{T}}_n}\Vert f-t{\Vert }_{\infty }$ and $n/m\le M$, it becomesThis is a near-best uniform approximation estimate; it follows from reproduction and boundedness without requiring the de la Vallée Poussin sum itself to be a best approximant.

For any degree-at-most-$n$ trigonometric polynomial $t_n$, linearity and reproduction giveThe operator norm bound in the preceding part yieldsTake the infimum over all such trigonometric polynomials. By the definition of best uniform approximation,No choice of a minimizer is needed for this argument. It is an instance of the polynomial reproduction error bound: a bounded linear reproducing operator has error at most $1+\Vert P\Vert$ times the optimal error.

The Runge phenomenon is the failure of increasing-degree polynomial interpolation at equally spaced interpolation nodes to converge to some smooth target functions, with large oscillations near the ends of the interval. It is a property of the approximation problem, and can occur even with exact arithmetic. A numerically stable evaluation method cannot remove it. Endpoint clustering, as in Chebyshev-Lobatto nodes, changes the approximation problem and can greatly improve it.^[ref]

Take $f(x)=1/(1+25x^2)$ on $[-1,1]$ and polynomial degree at most $10$. Use either $x_k=-1+k/5$ or $x_k=\cos (k\pi /10)$ for $k=0,\dots ,10$. Each interpolation polynomial passes through all eleven of its own samples, yet the first has severe endpoint oscillations and the second stays much closer to the target function. Evaluating both by the barycentric interpolation formula on $20,001$ equally spaced points, including the endpoints, gives sampled maximum absolute errors of approximately $1.915659$ and $0.132197$, respectively: about a $14.49$-fold reduction. These are maxima over the stated grid, not certified maxima over the continuous interval. The generator separately checks exact reproduction of all stored nodal values.

The polynomial interpolation error formula explains why interpolation nodes enter the analysis. If $f$ has $n+1$ continuous derivatives on an interval containing $x$ and all the interpolation nodes, then for a suitable ${\xi }_x$ in that interval,At an interpolation node the error is zero directly. Away from the interpolation nodes, both the product and the high-order derivative matter. A short proof applies Rolle's theorem repeatedly to $f(t)-p_n(t)-[f(x)-p_n(x)]{\prod }_i(t-x_i)/{\prod }_i(x-x_i)$, which vanishes at $n+2$ distinct points.^[ref]

The successful comparison here does not assert that Chebyshev-Lobatto nodes solve every approximation problem for every function. The derivatives, the polynomial degree, and measurement errors still matter. The Lebesgue constant of interpolation measures worst-case amplification of nodal errors, while the polynomial reproduction error bound relates interpolation to the best uniform approximation available at that polynomial degree.