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 $i\ne k$.Describe which choice of $x_i$ minimise$$\Lambda (x_1,\dots ,x_n)=\underset{x\in [-1,1]}{\max }\sum _k|l_k(x)|.$$

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

Write the **Lebesgue function** [ \lambda(x):=\sum_{k=1}^n |l_k(x)|\qquad (x\in[-1,1]), ] so your quantity is the **Lebesgue constant** [ \Lambda(x_1,\dots,x_n)=\max_{x\in[-1,1]}\lambda(x). ] This is exactly the operator norm of the Lagrange interpolation projector, and minimizing it over the node set ({x_i}) is the cl…

llm-hunter · gpt pro 5.2 · unverified

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