Lebesgue constant

Remarks

Let $I_n \colon C[a, b] \to C[a, b]$ send $f$ to its Lagrange interpolation polynomial $\sum_j f(x_j) \ell_j$. With the uniform norm, $\Lambda_n$ is the operator norm of $I_n$.

Example: nodes $-1, 0, 1$ in $[-1, 1]$. Here $\ell_0 = x(x - 1)/2$, $\ell_1 = 1 - x^2$ and $\ell_2 = x(x + 1)/2$. For $0 \le x \le 1$ this gives $\lambda_2(x) = 1 + x - x^2$, and by symmetry $\Lambda_2 = \lambda_2(1/2) = 5/4$.

Non-example: the number $\max_x \abs{\sum_j \ell_j(x)}$ is not the Lebesgue constant; it always equals $1$, because $\sum_j \ell_j = 1$. The absolute values must be taken term by term.

Always $\Lambda_n \ge 1$. For equally spaced nodes $\Lambda_n$ grows like $2^{n+1} / (e \, n \log n)$. For Chebyshev nodes $\Lambda_n = \frac{2}{\pi} \log n + O(1)$, and no choice of nodes does better than $\frac{2}{\pi} \log n - C$ for an absolute constant $C$.