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

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## gpt-5-2-pro[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=1 "Edit section: gpt-5-2-pro") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=1 "Edit section's source code: 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…[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=2 "Edit section: 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…") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=2 "Edit section's source code: 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[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=3 "Edit section: Assistant: Write the Lebesgue function") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=3 "Edit section's source code: 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[[2]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-2) 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[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=4 "Edit section: 1) Is") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=4 "Edit section's source code: 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[[3]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-3) 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[[4]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-4) 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[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=5 "Edit section: 2) Which") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=5 "Edit section's source code: 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 𝑋\*={𝑥 1\*<⋯<𝑥 𝑛\*}X\*={x 1\*​<⋯<x n\*​} is the (unique) node set for which 𝜆 0(𝑋\*)=𝜆 1(𝑋\*)=⋯=𝜆 𝑛(𝑋\*),λ 0​(X\*)=λ 1​(X\*)=⋯=λ n​(X\*),i.e. the 𝑛+1 n+1 “relative maxima” of Λ 𝑋\*Λ X\*​on the 𝑛+1 n+1 subintervals[𝑥 𝑖\*,𝑥 𝑖+1\*][x i\*​,x i+1\*​] all have the same height.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[[5]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-5) 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[[6]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-6)

##### What do these maximizing nodes look like?[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=6 "Edit section: What do these maximizing nodes look like?") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=6 "Edit section's source code: 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[[7]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-7)_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[[8]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-8) 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[[9]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-9)

#### Summary[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=7 "Edit section: Summary") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=7 "Edit section's source code: 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[[10]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-10)_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[[11]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-11)_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[[12]](https://freemwiki.com/index.php?oldid=595788&title=Openai%2F696b1675-eda0-800c-80f9-612b39635436#cite_note-12)

## References[[edit](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&veaction=edit&section=8 "Edit section: References") | [edit source](https://freemwiki.com/index.php?title=Openai/696b1675-eda0-800c-80f9-612b39635436&action=edit&section=8 "Edit section's source code: References")]

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