Volume 6 • Issue 1 • PP: 01–06 • 2026
Spectrally Matched Fractional Sobolev Tikhonov Regularization for Periodic Deconvolution
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© 2026 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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Let T = R/Z and Aβ = (I−Δ)−β/2 with β > 0. From data gδ = Aβ f †+η, ∥η∥2 ≤ δ, consider the fractional Sobolev Tikhonov family f δ λ,s = argminf∈Hs(T) Aβ f −gδ 2 2 +λ ∥ f ∥2 Hs, s ≥ 0. Writing μk = (1+4π2k2)1/2 and ρ = s+β, the estimator is diagonal in the Fourier basis, bf δ λ,s(k) = μβ k 1+λ μ2ρ k bgδ (k). If f † ∈ Hr(T) and 0 < r ≤ 2ρ, then f δ λ,s− f † 2 ≤Caδλ−a+CbMrλb, a = β 2ρ , b = r 2ρ , Mr = f † Hr . where Cq = qq(1−q)1−q for 0 < q < 1 and C1 = 1. Direct minimization gives λ∗ ≍ δ2ρ (r+β) and f δ λ∗,s− f † 2 = O δr/(r+β). Thus the pre-saturation exponent is independent of the penalty order s. When r > 2ρ, the bias saturates and the rate becomes O(δ2ρ/(2ρ+β)); hence the least order avoiding saturation is smin = max{0, r/2−β}. The balanced half-power frequency satisfies μc ≍ δ−1/(r+β), whereas its logarithmic roll-off is −ρ/2. The same exponents persist for spectrally equivalent convolution operators c−μ−β k ≤ ak ≤ c+μ−β k . Fourier experiments on N = 2048 modes, using three source regularities, four penalty orders, five noise levels, and 30 perturbations per configuration, reproduce the predicted saturation ordering and transition boundary. For the H6 benchmark at δ = 10−3, the mean L2 error decreases from 1.4013×10−2 for s = 0 to 1.2710×10−3 for s = 2; the latter is the first tested order in the non-saturated regime.
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