Openai/696b1675-eda0-800c-80f9-612b39635436

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gpt-5-2-pro

User: For $x1,\\ldots,xn\\in -1,1$ let\\lkx\\frac{\\prod{i\\neq k}x-xi}{\\prod{i\\neq k}xk-xi},\\which are such that $lkxk1…

For $x_1,\\ldots,x_n\\in [-1,1]$ let\\[l_k(x)=\\frac{\\prod_{i\\neq k}(x-x_i)}{\\prod_{i\\neq k}(x_k-x_i)},\\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\\neq k$.\n\nLet $x_0=-1$ and $x_{n+1}=1$ and\\[\\Upsilon(x_1,\\ldots,x_n)=\\min_{0\\leq i\\leq n}\\max_{x\\in[x_i,x_{i+1}]} \\sum_k \\lvert l_k(x)\\rvert.\\]Is it true that\\[\\Upsilon(x_1,\\ldots,x_n)\\ll \\log n?\\]Describe which choice of $x_i$ maximise $\\Upsilon(x_1,\\ldots,x_n)$.

Assistant: Write the Lebesgue function

Write the Lebesgue function

ΛX(x)  :=  ∑k=1n∣lk(x)∣,X={x1<⋯<xn}⊂(−1,1),\Lambda_X(x)\;:=\;\sum_{k=1}^n |l_k(x)|,\qquad X=\{x_1<\cdots<x_n\}\subset(-1,1),ΛX​(x):=k=1∑n​∣lk​(x)∣,X={x1​<⋯<xn​}⊂(−1,1), and extend the partition points by x0=−1, xn+1=1x_0=-1,\ x_{n+1}=1x0​=−1, xn+1​=1. A basic (and very useful) fact is that ΛX\Lambda_XΛX​ is piecewise polynomial and, on each subinterval (xi,xi+1)(x_i,x_{i+1})(xi​,xi+1​), it has exactly one local maximum; if ±1\pm1±1 are not interpolation points, there are also local maxima “at the endpoints” ±1\pm1±1. DNB Portal So if we set

λi(X)  :=  max⁡x∈[xi,xi+1]ΛX(x)(i=0,1,…,n),\lambda_i(X)\;:=\;\max_{x\in[x_i,x_{i+1}]}\Lambda_X(x)\qquad (i=0,1,\dots,n),λi​(X):=x∈[xi​,xi+1​]max​ΛX​(x)(i=0,1,…,n), then your quantity is simply the smallest local maximum

Υ(X)=min⁡0≤i≤nλi(X).\Upsilon(X)=\min_{0\le i\le n}\lambda_i(X).Υ(X)=0≤i≤nmin​λi​(X).

1) Is

Υ ( 𝑋 ) ≪ log ⁡ 𝑛 Υ(X)≪logn for all 𝑋 X?

Yes — and the right “reason” is that the max–min problem you wrote down is already a classical extremal problem in interpolation theory (Bernstein–Erdős). One formulation (in modern notation) is:

• Let M(X):=max⁡iλi(X)=max⁡x∈[−1,1]ΛX(x)M(X):=\max_i \lambda_i(X)=\max_{x\in[-1,1]}\Lambda_X(x)M(X):=maxi​λi​(X)=maxx∈[−1,1]​ΛX​(x) be the usual Lebesgue constant.
• Let m(X):=min⁡iλi(X)=Υ(X)m(X):=\min_i \lambda_i(X)=\Upsilon(X)m(X):=mini​λi​(X)=Υ(X) be the smallest local maximum.

Then there exists a distinguished point set X\X^\X\* (“Lebesgue–optimal nodes”) for which the local maxima are all equal (equioscillation), and for every node set XXX one has the sandwich inequality

m(X) ≤ m(X\) = M(X\) ≤ M(X),m(X)\ \le\ m(X^\)\ =\ M(X^\)\ \le\ M(X),m(X) ≤ m(X\) = M(X\) ≤ M(X), i.e. the same set X\X^\X\* maximizes the smallest local maximum and minimizes the largest one. DNB Portal Equivalently,

Υ(X)≤Υ(X\)=M(X\).\Upsilon(X)\le \Upsilon(X^\) = M(X^\).Υ(X)≤Υ(X\)=M(X\). Now M(X\)M(X^\)M(X\*) is the optimal Lebesgue constant, and it is known to have logarithmic growth:

M(X\)  =  2π(log⁡(n+1)+γ+log⁡4π)+o(1)(n→∞),M(X^\) \;=\; \frac{2}{\pi}\Big(\log(n+1)+\gamma+\log\frac{4}{\pi}\Big)+o(1) \qquad (n\to\infty),M(X\*)=π2​(log(n+1)+γ+logπ4​)+o(1)(n→∞), in the standard indexing conventions (shifting n↦n−1n\mapsto n-1n↦n−1 just changes log⁡(n+1)\log(n+1)log(n+1) to log⁡n\log nlogn at this level). DNB Portal Therefore,

Υ(X) ≤ Clog⁡n\Upsilon(X)\ \le\ C\log nΥ(X) ≤ Clogn for an absolute constant CCC, uniformly over all XXX.

Moreover, this is sharp in order: at the extremal set X\X^\X\, Υ(X\)∼2πlog⁡n\Upsilon(X^\)\sim \tfrac{2}{\pi}\log nΥ(X\)∼π2​logn.

2) Which

𝑥 𝑖 x i ​

maximize 

Υ Υ?

They are exactly the Lebesgue–optimal interpolation nodes X\X^\X\*, characterized by an equioscillation/equal–peak property:

The maximizer

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DNB Portal

+1

This is precisely the Bernstein–Erdős characterization: the optimal set is unique, symmetric (about 0), and its Lebesgue function must equioscillate. DNB Portal In general the points xi\x_i^\xi\*​ do not have a simple closed form; explicit formulas are only known for very small degrees, and otherwise they are computed numerically (e.g. via a Remez-type exchange algorithm). DNB Portal

What do these maximizing nodes look like?

While not closed-form, their shape is well understood: They are symmetric: xn+1−k\=−xk\x_{n+1-k}^\=-x_k^\xn+1−k\​=−xk\*​. DNB Portal They cluster toward ±1\pm1±1 with an asymptotic arcsine/Chebyshev density, i.e. heuristically xk\≈cos⁡ ⁣((2k−1)π2n),x_k^\\approx \cos\!\Big(\frac{(2k-1)\pi}{2n}\Big),xk\​≈cos(2n(2k−1)π​), (Chebyshev–Gauss nodes), with small nnn-dependent corrections. DNB Portal There are also explicit near-optimal “modified Chebyshev” node families in the literature that achieve the same asymptotic 2πlog⁡n\frac{2}{\pi}\log nπ2​logn growth. DNB Portal

Summary

Yes: for every choice of nodes, Υ(x1,…,xn)=O(log⁡n)\Upsilon(x_1,\dots,x_n)=O(\log n)Υ(x1​,…,xn​)=O(logn). In fact Υ(X)≤M(X\)∼2πlog⁡n\Upsilon(X)\le M(X^\)\sim \frac{2}{\pi}\log nΥ(X)≤M(X\)∼π2​logn. DNB Portal The node set that maximizes Υ\UpsilonΥ is the unique Lebesgue–optimal set X\X^\X\, characterized by equal local maxima (equioscillation) of the Lebesgue function on all n+1n+1n+1 subintervals. DNB Portal Asymptotically, those points are distributed like Chebyshev points (arcsine law), so xk\x_k^\xk\​ are very close to cos⁡((2k−1)π/(2n))\cos((2k-1)\pi/(2n))cos((2k−1)π/(2n)) for large nnn. DNB Portal