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

==== 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<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>