The Wallis Weight Kernel: A Family of Soft-Cutoff Regularization Weights from the Classical Wallis Product

In this paper, I introduce a family of Wallis Weight Kernels: Wₛ (n) = ∏₊=₁ⁿ (2k-s) / (2k), for s ∈ ℝ. For s = 1, this reduces to the classical Wallis product partial sequence W (n) ~ 1/√ (π n). I prove the asymptotic behavior of the general family, demonstrate its utility in regularizing higher-order harmonic series, and provide closed-form evaluations: ∑ W (n) /n = 2 ln 2∑ W (n) /n² = π²/4 - 2 ln² 2 I compare the kernel to standard regularization methods (Zeta, Hard Cutoff, Borel) and show its advantages as a smooth, algebraically natural UV cutoff. This work provides a foundational tool for the Unified Regularization Principle. Key results: - A parameterized family of Wallis kernels- Soft-cutoff UV regularization weight- Closed-form evaluations for harmonic series- Comparison with existing regularization methods