Local spectral density and Slepian concentration for spherical Fourier-Bessel truncation spaces

Abstract

We study diagonal kernel asymptotics and concentration spectra for a family of non-translation-invariant spectral projections in ${\mathbb{R}}^d$, $d\ge 2$. 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 $n\le N$ 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 $N/K\to \hat{I}º$, 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 $W_{⟨\hat{I}º⟩}^{⟨d⟩}(\Vert x\Vert )=W^{⟨d⟩}(\Vert x\Vert /\hat{I}º)$. Its constant plateau recovers the constant density of the classical Paley--Wiener projection for $\Vert x\Vert \lesssim \hat{I}º$, while its far-field tail, when weighted by the spherical volume element, yields a Hankel-type radial density law with angular-bandwidth factor $\hat{I}{º}^{d-1}$. Thus the classical Paley--Wiener concentration problem is recovered at the endpoint $\hat{I}º=\infty$, whereas finite $\hat{I}º$ exhibits a transition from a Fourier-like local-density region to a Hankel-type radial-density regime, with $\hat{I}º$ 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 $\hat{I}º$-dependent density over the localization domain.

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