Erdős Problem #1152

The question

For $n\ge 1$ fix some sequence of $n$ distinct numbers $x_{1n},\dots ,x_{nn}\in [-1,1]$. Let $ϵ=ϵ(n)\to 0$.

Does there always exist a continuous function $f:[-1,1]\to \mathbb{R}$ such that if $p_n$ is a sequence of polynomials, with degrees $\mathrm{deg}{p}_n<(1+ϵ(n))n$, such that $p_n(x_{kn})=f(x_{kn})$ for all $1\le k\le n$, then $p_n(x)\nrightarrow f(x)$ for almost all $x\in [-1,1]$?

This is Problem 2.42 in [Va99]. The $n$ nodes of the $n$-th row form the finite set nodes n. We take $ϵ(n)>0$. A smaller $ϵ$ only shrinks the class of $p_n$, so this is equivalent to allowing any real $ϵ(n)\to 0$. We impose the degree bound only for $n\ge 1$, since for $n=0$ the bound natDegree < 0 can never hold. We read "$p_n(x)\nrightarrow f(x)$ for almost all $x$" as: for almost every $x$, the sequence $p_n(x)$ does not converge to $f(x)$.

For $\mathrm{deg}{p}_n<n$ the interpolating polynomial is unique (the Lagrange polynomial). Erdős, Kroó and Szabados [EKS89, pp. 232-233] give necessary and sufficient conditions on the nodes under which, for every continuous $f$ and every fixed $ϵ>0$, there are interpolating polynomials of degree at most $\lfloor n(1+ϵ)\rfloor$ approximating $f$ at the rate of the best uniform approximation (in particular converging uniformly to $f$).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1152».

theorem erdos_1152 :
    ∀ nodes : ℕ → Finset ℝ, (∀ n, (nodes n).card = n) →
      (∀ n, ∀ x ∈ nodes n, x ∈ Set.Icc (-1 : ℝ) 1) →
      ∀ ε : ℕ → ℝ, (∀ n, 0 < ε n) → Tendsto ε atTop (𝓝 0) →
        ∃ f : ℝ → ℝ, ContinuousOn f (Set.Icc (-1 : ℝ) 1) ∧
          ∀ p : ℕ → ℝ[X],
            (∀ n : ℕ, 1 ≤ n → ((p n).natDegree : ℝ) < (1 + ε n) * n) →
            (∀ n, ∀ x ∈ nodes n, (p n).eval x = f x) →
              ∀ᵐ t ∂(volume.restrict (Set.Icc (-1 : ℝ) 1)),
                ¬ Tendsto (fun n => (p n).eval t) atTop (𝓝 (f t))

What counts as progress

• A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1152. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• erdosproblems.com/1152
• [EKS89] Erdős, P., Kroó, A. and Szabados, J., On convergent interpolatory polynomials. J. Approx. Theory 58 (1989), 232-241. PDF
• [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.