Wavelet covariances under fixed finite observations: dilation symmetry breaking and distinct endpoint limits

Abstract

For a centered homogeneous generalized Gaussian field with spectral weight $C|ω|^{-γ}$, we prove two-endpoint recovery of normalized posterior wavelet covariances under fixed finite admissible observations. Global dilation symmetry holds if and only if all observation functions vanish. The nonzero wavelet and the observation functions are real Schwartz functions with $M\geq1$ vanishing moments, with $C>0$ and $γ<2M+1$. Noise is Gaussian and field-independent, and total observation covariance is positive definite. Under simultaneous dilation of target scales and positions, multiplication by $λ^{-γ}$ recovers the prior covariance as $λ\downarrow0$ and $λ\uparrow\infty$, uniformly on every fixed compact target set. If a nonzero observation function exists, additional finite first nonzero Fourier orders $m_ψ,m_i$ at zero of the wavelet and nonzero observation functions yield compact-uniform recovery bounds $O(λ^{2m_ψ+1-γ})$ and $O(λ^{-(2\min_i m_i+1-γ)})$, with leading coefficients and cancellation-dependent improvements. The bounds are exact leading orders when their coefficients are nonzero. For the Matérn-type spectrum $S(ω)=C(κ^2+ω^2)^{-p}$, $κ>0$, $1/2<p<M+1/2$, fixed finite noisy point observations yield distinct compact-uniform limits: the homogeneous $C|ω|^{-2p}$ kernel after multiplication by $λ^{-2p}$ at small scales, and the constant-spectrum $S(0)$ kernel without normalization at large scales. A projection-norm criterion treats fixed closed observation subspaces; Mexican-hat formulas illustrate the finite-observation results.