2024/02/27 by Xinpeng Huang, Huang, Xinpeng
Mathematics · #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2402.17444
openalex publication_date 2024/02/27 · openalex created_date 2024/03/05 · openalex updated_date 2026/07/28
We study diagonal kernel asymptotics and concentration spectra for a family of non-translation-invariant spectral projections in ℝd, d≥ 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≤ 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→κ, we identify the limiting profile of the normalized diagonal reproducing kernel K-dKN,K(x,x), interpreted as the local density encoded by the SFB projection. The profile is the rescaled radial transition function W⟨ d⟩⟨κ⟩(‖ x‖)=W⟨ d⟩(‖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 κd-1. Thus the classical Paley--Wiener concentration problem is recovered at the endpoint κ=∞, 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.