Localized frames on Euclidean balls

Abstract

We construct explicit wave packet frames adapted to Euclidean balls and use them to obtain quantitative eigenvalue estimates for spatio--spectral limiting operators. Let \(d\geq 2\), let \(B_d(R)\subset \R^d\) be the Euclidean ball of radius \(R\), and let \(S\subset \R^d\) be a measurable set such that $\partial S$ has finite $(d-η)$-upper Minkowski content for $0 < η\leq 1$. We construct a unit-norm frame for \(L^2(B_d(R))\), with frame bounds depending only on the dimension $d$, whose elements are adapted to the radial and angular geometry of the ball. We prove quantitative Fourier localization estimates for this frame: Relative to \(S\), the frame decomposes into packets concentrated in \(S\), packets concentrated in \(\R^d\setminus S\), and an exceptional family whose cardinality is bounded explicitly in terms of \(R\), and the Minkowski content of \(\partial S\). As an application, we derive an upper bound for the plunge region of the spatio-spectral limiting operator associated to the sets $B_d(R)$ and $S$.