Erdős Problem #671

Erdős problem #671 · erdosproblems.com/671

For triangular arrays of nodes $a_i^n\in[-1,1]$ let $\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$, with fundamental polynomials $p_i^n$. Is there a choice of nodes such that for every continuous $f$ there is some $x$ where $\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty$ and yet $\mathcal{L}^nf(x) \to f(x)$? Is there a choice with $\limsup_n \sum_i\lvert p_{i}^n(x)\rvert=\infty$ for every $x$, yet for every continuous $f$ some $x$ has $\mathcal{L}^nf(x)\to f(x)$? Both questions are claimed resolved in the affirmative.

Result

Proved(see note)

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Construction

Field

Analysis, Interpolation

Posed by

Paul Erdős

Year posed

1982

Years open

44y

Solved

2026-06-22

Model

GPT-5.5 Pro, Codex

Vendor

OpenAI

Collaborators

Liam Price

Verification

Lean-verified

Publication

Announced

Significance

10 / 100*

Disclosed cost

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Wikipedia

No dedicated article