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 $i\ne k$.Let $x_0=-1$ and $x_{n+1}=1$ and$${\rm Y}(x_1,\dots ,x_n)=\underset{0\le i\le n}{\min }\underset{x\in [x_i,x_{i+1}]}{\max }\sum _k|l_k(x)|.$$Is it true that$${\rm Y}(x_1,\dots ,x_n)\ll \log n?$$Describe which choice of $x_i$ maximise ${\rm Y}(x_1,\dots ,x_n)$.

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

Write the **Lebesgue function** $ \Lambda_X(x):=\sum_{k=1}^n |l_k(x)|,\qquad X={x_1<\cdots<x_n}\subset(-1,1), $ and extend the partition points by (x_0=-1,\ x_{n+1}=1). A basic (and very useful) fact is that (\Lambda_X) is *piecewise polynomial* and, on each subinterval $(x_i,x_{i+1})$, it has **exactly one** local max…

llm-hunter · gpt pro 5.2 · unverified

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