Boostlets for cone-concentrated wavefields

Abstract

Abstract

Abstract Wavelets localize position and scale, but the nondispersive wave equation organizes high-frequency energy on characteristic cones. This Letter gives a quantitative obstruction to using ordinary isotropic wavelets as atoms of cone-concentrated propagation and shows how it is removed by adding Lorentz rapidity. The analysis is a finite-window microlocal benchmark for band-limited packets near one regular branch of the cone. In light-cone frequency variables, a normalized wave packet with transverse cone width $\eps$ has covariance $\diag(1,\eps^2)$. Its largest possible squared overlap with any isotropically dilated Gaussian wavelet is exactly $4\eps/(1+\eps)^2$, whereas a boosted wavelet, or boostlet, with rapidity $\theta=\frac12\log \eps^{-1}$ and scale $a=\eps^{-1/2}$ matches the covariance and attains unit overlap, up to the exponentially small admissibility correction of a Morlet window. Hence an isotropic wavelet expansion has entropy at least $\log[(1+\eps)^2/(4\eps)]$ and requires $\Omega(\eps^{-1})$ coefficients to capture fixed energy, while one boostlet coefficient suffices in the local Gaussian model. A weak quartic cone deformation produces a boostlet sparsity defect proportional to $\lambda^2K^6/\eps^2$ under the stated perturbative scale assumption.