Averaged Cauchy products of convergent sequences

DefinitionThe claim of Scottish Book Problem 8#✓fully proved

This is the statement, asked about in form (b) of Mazur's Problem 8 in the Scottish Book typescript, that every convergent sequence \((z_n)_{n \ge 0}\) of real numbers can be written as \(z_n = \frac{1}{n+1} \sum_{k=0}^{n} x_k y_{n-k}\) for all \(n \ge 0\), where \((x_n)_{n \ge 0}\) and \((y_n)_{n \ge 0}\) are convergent sequences of real numbers.

Indices are shifted to start at \(0\).

def ScottishBookProblem8Claim : Prop :=
  ∀ (z : ℕ → ℝ), (∃ L, Filter.Tendsto z Filter.atTop (nhds L)) →
    ∃ (x y : ℕ → ℝ), (∃ Lx, Filter.Tendsto x Filter.atTop (nhds Lx)) ∧
      (∃ Ly, Filter.Tendsto y Filter.atTop (nhds Ly)) ∧
      ∀ n, z n = (∑ k ∈ Finset.range (n + 1), x k * y (n - k)) / (n + 1 : ℝ)

Used by 1

Used byMazur's Problem 8 has a negative answer

TheoremMazur's Problem 8 has a negative answer#●ready to prove

There is a convergent sequence \((z_n)_{n \ge 0}\) of real numbers that cannot be written as \(z_n = \frac{1}{n+1} \sum_{k=0}^{n} x_k y_{n-k}\) for all \(n \ge 0\) with convergent sequences \((x_n)_{n \ge 0}\) and \((y_n)_{n \ge 0}\) of real numbers. That is, the claim of Scottish Book Problem 8 is false.

theorem_wanted mazur8CauchyProduct_refuted : ¬ ScottishBookProblem8Claim

Sources

• S. Mazur, Problem 8 of the Scottish Book; negative solution by V. V. Peller, "On S. Mazur's Problems 8 and 88 from the Scottish Book", arXiv:math/0703716, building on Peller, J. Operator Theory 7 (1982) 341-372; independently Kwapien-Pelczynski and Eggermont-Leung, Proc. Amer. Math. Soc. 96 (1986) 269-274.

Uses 1

Statement usesThe claim of Scottish Book Problem 8

1 fully proved1 ready to prove

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