Localized frames on Euclidean balls

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 $(d-\eta)$-upper Minkowski content for $0 < \eta \leq 1$. 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 $S$.