The sharp shape-parameter law for sinh and Kaiser-Bessel regularised Shannon sampling, with a corrected alias-inclusive error bound

Unrefereed research candidate. Windowed (regularised) Shannon sampling rebuilds a bandlimited signal from its 2m nearest samples by multiplying the cardinal sine by a compactly supported window; the sinh-type and continuous Kaiser–Bessel (cKB) windows, which are also NFFT and gridding kernels, each have a shape parameter β. For angular bandwidth δ < π put d = π − δ. Kircheis, Potts and Tasche (BIT Numer. Math. 65, 2025) conjectured that β = dm gives the fastest exponential error decay.

This package proves that the worst-case error over the L² unit ball satisfies log E = −β + √((β² − d²m²)₊) + O(log m) uniformly in β, for both windows, so the optimal exponential rate is d and is attained exactly when β/m → d (the worst-case reading of the conjectures). Under-tuning costs accuracy linearly, over-tuning like a square root; exact optimality of dm at finite m fails. The same law holds for each signal whose spectrum has bounded variation and is bounded away from zero. Leading constants, a second-order sinh tuning law and family minima are given. The general error estimate used for these windows since 2022 omits aliased bands: an exact rational certificate violates it by a factor of about 3500. An alias-inclusive replacement is proved for a narrower window class containing sinh and cKB, and the published explicit sinh and cKB bounds are shown to remain true. Optimal pointwise linear recovery from the same samples has root-limsup rate tan²(δ/4), a larger exponent than either window attains. Certified interval-arithmetic tables quantify the finite-m picture for m ≤ 32. Conjecture 9.7 (cKB oscillation) and the cKB onset law are conjectural or exploratory.

The archive contains the 45-page manuscript (PDF and LaTeX), Python scripts (Arb ball arithmetic via python-flint; exact rational arithmetic for the counterexample), JSON certificates, a one-command byte-for-byte replay, semantic negative controls and SHA-256 sums. Assurance: written proofs; certified computations; producer-side replay of all 18 outputs, byte-identical, including a clean replay from a separate copy; internal AI-assisted review, an AI-assisted adversarial review by a different model and two targeted proof audits. This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.