For $x_1,\dots ,x_n\in [-1,1]$ let$$l_k(x)=\frac{{\prod }_{i\ne k}(x-x_i)}{{\prod }_{i\ne k}(x_k-x_i)},$$which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\ne k$.What is the minimal value of$$I(x_1,\dots ,x_n)={\int }_{-1}^1\sum _k|l_k(x){|}^2\mathrm{d}x?$$In particular, is it true that$$\min I=2-(1+o(1))\frac{1}{n}?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (P_{n-1}) be the space of real polynomials of degree (\le n-1), and let $\\{\phi_j\\}_{j=0}^{n-1}$ be any orthonormal basis of (L^2([-1,1])\cap P*{n-1}) (e.g. normalized Legendre polynomials).

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.