Numerical Verification of Erdős Problem #1131 via Discrete Christoffel Functions

This notebook presents a reproducible numerical study of \emph{Erd\H{o}s Problem \#1131}, concerning the asymptotic behavior of the discrete Christoffel function arising from Lagrange interpolation on $[-1,1]$.

The numerical framework is based on Legendre polynomials and the Christoffel--Darboux kernel. Interpolation nodes are chosen as the zeros of the \emph{Legendre--integral polynomial}
\[
Q_n(x) = \frac{P_n(x) - P_{n-2}(x)}{2n-1},
\]
where $P_n$ denotes the Legendre polynomial of degree $n$. Using these nodes, the notebook evaluates the diagonal Christoffel--Darboux kernel
\[
K_n(x,x) = \sum_{k=0}^n \frac{2k+1}{2} P_k(x)^2
\]
and the corresponding discrete Christoffel function in interpolation normalization,
\[
\lambda_n(x) = \left( \frac{1}{n} \sum_{k=1}^n \frac{K_{n-1}(x,x)}{K_{n-1}(x_k,x_k)} \right)^{-1}.
\]

The Erd\H{o}s functional
\[
I_n = \int_{-1}^1 \sum_{k=1}^n |l_k(x)|^2 \, dx
\]
is computed numerically using Gauss--Legendre quadrature, and its convergence toward the conjectured limit $I_n \to 2$ is investigated through the scaled quantity $n(2-I_n)$.

In addition, the notebook explores the endpoint asymptotic regime by plotting the scaled discrete Christoffel function
\[
n^{5/3}\lambda_n\!\left(1 - \frac{u}{n^{2/3}}\right),
\]
providing numerical evidence of Airy-type behavior near the boundary of the interval.

All computations are implemented in Python using \texttt{NumPy} and \texttt{Matplotlib}, and the notebook is fully compatible with Google Colab. The results offer numerical support for conjectures related to Erd\H{o}s Problem \#1131 and contribute computational evidence to questions in approximation theory and orthogonal polynomials.