Comparison between exact kernels (Eq. (53) of Hanasoge 2018) in solid navy blue lines and asymptotic kernels (𝒦_nℓ(r) multiplied by f_{ℓ′−ℓ,s}) in red-dashed lines for s = 6. The panels on the left and right sides are for ℓ′−ℓ = 1 and 3, respectively. We consider three different harmonic degrees ℓ = 50, 100, 150 which cover almost the entire range of harmonic degrees that we have used for data preparation. We have normalized both kernels in each panel by the maximum values of the kernel obtained from the exact expression. Both kernels match better at larger values of ℓ, as expected.