theorem Chapter3.problem_3_5_sobolev_summable {M : ℕ} {β Q : ℝ} (hβ : 1 / 2 < β) (hQ : 0 < Q) (θ : Fin M → ℝ) (hθ : θ ∈ SobolevEllipsoid β Q M) : ∑ j : Fin M, |θ j| ≤ √Q * √(∑ j : Fin M, if sobolevCoeff β (↑j + 1) = 0 then 0 else (sobolevCoeff β (↑j + 1))⁻¹ ^ 2)

Problem 3.5 — Sobolev regularity β > 1/2 implies absolute summability.

Rigollet, High-Dimensional Statistics, Problem 3.5 (line 2584).

If f ∈ Θ(β, Q) for β > 1/2 and Q > 0, then f is continuous. The key ingredient is showing that the Fourier coefficients θⱼ of f are absolutely summable: Σⱼ |θⱼ| < ∞.

The proof applies the Cauchy–Schwarz inequality: Σ |θⱼ| = Σ (aⱼ |θⱼ|) · aⱼ⁻¹ ≤ (Σ aⱼ² θⱼ²)^{1/2} · (Σ aⱼ⁻²)^{1/2} ≤ √Q · (Σ aⱼ⁻²)^{1/2} where aⱼ = j^β are the Sobolev weights. The series Σ j^{−2β} converges precisely when 2β > 1, i.e. β > 1/2. Absolute summability of the Fourier coefficients then gives uniform convergence of the Fourier series, hence continuity of f.