Certified Hermite transmutation of C-regularized cosine functions: optimal scaling, intrinsic conditioning, and reconstruction from noisy heat snapshots

The Hermite expansion of an exponentially bounded $C$-regularized cosinefunction (Ameziane Hassani et al., 2024) reconstructs the whole solutionfamily of $u''=Wu$ from a single snapshot of the associated heat semigroup:a transmutation from diffusion to waves. We develop its complete numericalanalysis. First, the snapshot time is a free parameter of the expansion;the resulting family removes the Gaussian time factor $e^{t^{2}/2}$ of allprevious error bounds, replacing it by a polynomial one, and yields explicitcertificates whose truncation order is linear in the phase$\Phi=t\sqrt{\norm{W}}$. Second, the full expansion applied to a perturbedsnapshot returns the exact solution plus the perturbation propagated by thetransmutation operator, whose norm $\kappa(s,t)$ is the intrinsic conditionnumber of the problem, and no algorithm can do better than$\eta\,e^{s\omega^{2}}\abs{\cos\omega t}$ on data of accuracy $\eta$. Thefloating-point rule $\norm{W}\lesssim4\abs{\ln\eps}$ is thereby identifiedas the resolution limit of the data, not a defect of the scheme, and thenoise-optimal snapshot time yields the universal accuracy law$\eps\,e^{\Phi/(2\sqrt2)}$, confirmed over fourteen orders of magnitude onthree meshes. Third, a certified error decomposition for noisy snapshotsmakes the truncation order a regularization parameter with an explicit apriori choice. Fourth, the Cauchy problem for the Laplace equation istreated as a $C$-regularized cosine function with a Gaussian mollifier,outside the scope of the original theorem. A comparison with theChebyshev--Bessel expansion delineates the scope of the method; an openPython implementation reproduces every figure.