Can Lagrange interpolation converge while the Lebesgue function diverges? (Erdős #671)
Statement
For each $n$ choose interpolation nodes $a_1^n,\ldots,a_n^n\in[-1,1]$, and let $p_i^n$ be the fundamental Lagrange polynomial of degree $n-1$ with $p_i^n(a_i^n)=1$ and $p_i^n(a_{i'}^n)=0$ for $i'\neq i$. For continuous $f:[-1,1]\to\mathbb{R}$ write $$\mathcal{L}^n f(x)=\sum_{1\le i\le n} f(a_i^n)\,p_i^n(x)$$ for the degree-$(n-1)$ interpolant of $f$ at these nodes, and call $\sum_{1\le i\le n}\lvert p_i^n(x)\rvert$ the Lebesgue function. Two questions. (1) Is there a choice of nodes $(a_i^n)$ such that for every continuous $f$ there exists some $x\in[-1,1]$ with $$\limsup_{n\to\infty}\sum_{1\le i\le n}\lvert p_i^n(x)\rvert=\infty\quad\text{and yet}\quad \mathcal{L}^n f(x)\to f(x)?$$ (2) Is there a choice of nodes such that $\limsup_{n\to\infty}\sum_{1\le i\le n}\lvert p_i^n(x)\rvert=\infty$ for every $x\in[-1,1]$, and yet for every continuous $f$ there exists $x\in[-1,1]$ with $\mathcal{L}^n f(x)\to f(x)$?
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable in Lean/Coq preferred, otherwise a full written proof) answering questions (1) and (2) — either an explicit node array $(a_i^n)$ with a proof that its Lebesgue function diverges as specified AND that $\mathcal{L}^n f$ converges at the required point(s) for every continuous $f$, or a proof that no such array exists. ADVANCES: settle exactly one of (1) or (2) with proof; or construct a node array realising the divergence side together with a proof of pointwise convergence for a restricted class of $f$ (e.g. Lipschitz, or real-analytic); or prove impossibility under an added structural constraint on the nodes. Deliver the node-array construction with its convergence/divergence proofs, or the impossibility proof.
Background
Posed by Erdős [Er82e, Er97f] (listed as Va99, 2.40), who offered a prize of $250. The Lebesgue function $\sum_i\lvert p_i^n(x)\rvert$ controls the interpolation operator's norm, so its unboundedness normally signals divergence — the questions ask whether pointwise convergence can nonetheless survive. Known obstructions: Bernstein [Be31] proved that for ANY node array there is a point $x_0\in[-1,1]$ where $\limsup_n\sum_i\lvert p_i^n(x_0)\rvert=\infty$ (the Lebesgue function is unbounded somewhere); Erdős and Vértesi [ErVe80] proved that for ANY node array there is a continuous $f$ with $\limsup_n\lvert\mathcal{L}^n f(x)\rvert=\infty$ for almost every $x$. Against this backdrop the problem probes whether a cleverly chosen node array can force convergence at at least one point for every $f$, despite an everywhere-unbounded Lebesgue function. Listed as open on erdosproblems.com/671 (fetched 2026-07-21, status 'open'). CAVEAT FOR THE REVIEWER: the page records one user-submitted 'claimed proof' in its forum thread, not incorporated into Bloom's curated remarks — its validity should be checked before posting. Distinct venue neighbours in the same interpolation family are Erdős #1132 (is $\limsup L_n(x)/\log n\ge 2/\pi$ a.e.?) and #1152 (vanishing degree-slack still forcing a.e. divergence). Attacker's tool: constructive node-array design (structured perturbations of Chebyshev nodes), quantitative Lebesgue-function and Lebesgue-constant estimates, Baire-category arguments over $C[-1,1]$, and numerical computation of Lebesgue functions and interpolation errors for candidate arrays to guide the construction.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #671 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.