Local spectral density and Slepian concentration for spherical Fourier-Bessel truncation spaces
We study diagonal kernel asymptotics and concentration spectra for a family of non-translation-invariant spectral projections in $\mathbb{R}^d$, $d\geq 2$. The projections are obtained from the classical Paley--Wiener projection by imposing, in spherical coordinates, an additional cutoff in the spherical-harmonic degree. Equivalently, they are the spherical Fourier--Bessel (SFB) truncation spaces in which, in addition to the radial Hankel/Bessel bandwidth $K$, only spherical harmonic degrees $n\leq N$ are retained. This angular cutoff preserves rotation invariance but breaks translation invariance, so the diagonal reproducing kernel has a spatially varying radial profile. In the coupled asymptotic regime $N/K\toκ$, we identify the limiting profile of the normalized diagonal reproducing kernel $K^{-d}\mathcal{K}_{N,K}(x,x)$, interpreted as the local density encoded by the SFB projection. The profile is the rescaled radial transition function $W^{\langle d\rangle}_{\langleκ\rangle}(\| x\|)=W^{\langle d\rangle}(\|x\|/κ)$. Its constant plateau recovers the constant density of the classical Paley--Wiener projection for $\|x\|\lesssimκ$, while its far-field tail, when weighted by the spherical volume element, yields a Hankel-type radial density law with angular-bandwidth factor $κ^{d-1}$. Thus the classical Paley--Wiener concentration problem is recovered at the endpoint $κ=\infty$, whereas finite $κ$ exhibits a transition from a Fourier-like local-density region to a Hankel-type radial-density regime, with $κ$ setting the radial scale of this transition. Using this local-density asymptotic, we prove an asymptotically bimodal eigenvalue distribution and a Shannon-number formula whose leading coefficient is the integral of this $κ$-dependent density over the localization domain.