Compressive Logarithmic Generating Elements for Inertialess Filtering of Symmetric Noise

We develop a polynomial inertialess filter for recovering a Gaussian signal from additive non-Gaussian noise within the Kunchenko decomposition framework, using a compressive sub-polynomial generating family. Following the mandatory linear element φ₁ = ξ, the family comprises the odd, monotone elements φᵢ = sign(ξ) log^(i−1)(1+|ξ|). These elements grow more slowly than any power of ξ, so they compress rather than amplify the tails; the normal system F^φ h = R^φ is finite whenever the noise has finite variance, without requiring moments above the second. The trade-off is numerical: successive powers of a slowly varying function are strongly collinear, making the system severely ill-conditioned and requiring Tikhonov regularisation with a cross-validated ridge. Monte Carlo experiments with 95% bootstrap confidence intervals, benchmarked against the Wiener linear filter through the Kunchenko coefficient e_(s,1), show that the logarithmic basis performs best on platykurtic and multimodal noise. At q = 0.5, e_(4,1) = 0.851; at q = 0.25, it decreases to 0.729, corresponding to a 27% reduction in residual variance relative to the linear filter. A controlled experiment with a real CWRU bearing carrier and a platykurtic disturbance reproduces the result (e_(4,1) = 0.853 for a bimodal disturbance at q = 0.5). Raw real noise with negative excess kurtosis alone provides no advantage, showing that the gain depends on disturbance structure rather than kurtosis alone. Because the generating elements are odd and sign-preserving, the mechanism is suited to symmetric non-Gaussian disturbances, particularly in the platykurtic and multimodal regime.