Editing Openai/696b1675-eda0-800c-80f9-612b39635436 (section)

=== 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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>
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]maxΛX(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≤nminλi(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):=maxiλi(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):=miniλi(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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>
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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>
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\'')∼π2logn.

==== 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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>  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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>

===== 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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>
'' 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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref> There are also explicit near-optimal “modified Chebyshev” node families in the literature that achieve the same asymptotic 2πlog⁡n\frac{2}{\pi}\log nπ2logn growth. DNB Portal<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>

==== 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\'')∼π2logn. DNB Portal<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>
'' 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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>
'' 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<ref>{{cite web|title=DNB Portal|url=https://d-nb.info/1090940718/34|publisher=DNB Portal|access-date=2026-01-18}}</ref>