Titchmarsh convolution theorem

DefinitionThe Volterra-form sub-conjecture of Scottish Book Problem 124#✓fully proved

The statement that for every continuous function \(f : \mathbb{R} \to \mathbb{R}\) with \(f(0) \ne 0\) and every function \(y : \mathbb{R} \to \mathbb{R}\) that is Lebesgue integrable on \([0, 1]\), if \(\int_0^x y(t) f(x - t)\,dt = 0\) for every \(x \in [0, 1]\), then \(y = 0\) almost everywhere on \([0, 1]\).

The Scottish Book typescript prints the upper limit of the integral as \(1\), but the entry's own arguments show that the intended upper limit is \(x\) (Volterra form); with upper limit \(1\) the claim is false, since \(f \equiv 1\) and \(y(t) = t - 1/2\) give a nonzero solution. The typescript does not specify a function class for \(y\); integrability on \([0, 1]\) is motivated by the \(L^2\) context of the entry.

def ScottishBookProblem124Claim : Prop :=
  ∀ (f : ℝ → ℝ), Continuous f → f 0 ≠ 0 →
    ∀ (y : ℝ → ℝ),
      MeasureTheory.IntegrableOn y (Set.Icc (0 : ℝ) 1) MeasureTheory.volume →
      (∀ x ∈ Set.Icc (0 : ℝ) 1, ∫ t in (0 : ℝ)..x, y t * f (x - t) = 0) →
      y =ᵐ[MeasureTheory.volume.restrict (Set.Icc (0 : ℝ) 1)] 0

Used by 1

Used byThe Volterra-form sub-conjecture of Scottish Book Problem 124 holds

TheoremThe Volterra-form sub-conjecture of Scottish Book Problem 124 holds#●ready to prove

For every continuous function \(f : \mathbb{R} \to \mathbb{R}\) with \(f(0) \ne 0\) and every function \(y : \mathbb{R} \to \mathbb{R}\) that is Lebesgue integrable on \([0, 1]\), if \(\int_0^x y(t) f(x - t)\,dt = 0\) for every \(x \in [0, 1]\), then \(y = 0\) almost everywhere on \([0, 1]\).

theorem_wanted marcinkiewicz124_holds : ScottishBookProblem124Claim

Sources

• Józef Marcinkiewicz, Problem 124 of the Scottish Book; typescript PDF p.58, printed p.55. TYPESCRIPT TYPO, CORRECTED (Phase 5): the typescript prints the equation as ∫₀¹ y(t) f(x-t) dt = 0, but the intended upper limit is x (Volterra form). The literal ∫₀¹ reading is false: f ≡ 1, y(t) = t - 1/2 is a nonzero solution. Only the Volterra ∫₀ˣ reading is filed here. Positive by the local Titchmarsh convolution theorem: E. C. Titchmarsh, "The zeros of certain integral functions", Proc. London Math. Soc. s2-25 (1926), 283–302, DOI 10.1112/plms/s2-25.1.283. Sketch: Titchmarsh gives λ, μ ≥ 0 with λ + μ ≥ 1 such that y = 0 a.e. on (0, λ) and f = 0 a.e. on (0, μ). Continuity of f at 0 with f 0 ≠ 0 forces μ = 0 (otherwise f would be a.e. zero on a positive-measure initial interval while being nonzero near 0), so λ ≥ 1 and y = 0 a.e. on [0, 1].

Uses 1

Statement usesThe Volterra-form sub-conjecture of Scottish Book Problem 124

1 fully proved1 ready to prove

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