Bessel-Conical Radial Basis

Integrate the basis at the input of a coordinate MLP, neural operator, or polar/spherical CNN patch. Normalize each sample to a known outer radius \(R\), compute \(\rho=\|x\|/(R+\varepsilon)\), and concatenate \(K\) radial features \(\phi_k(\rho)=J_b(j_{b,k}\rho)\) to Cartesian or Fourier features. Start with \(K=8\) and fixed \(b=\max(0,(n-2)/2)\). For a non-spherical cross-section, estimate the link eigenvalue \(\mu_1\) once using a finite-element or graph eigensolver, then set \(b=\sqrt{(n-2)^2/4+\mu_1}\). Compute Bessel zeros with a scientific-library root finder and cache them. Compare against learned sinusoidal radial features and no radial features at equal width. The theorem predicts that rescaling the domain from \(R\) to \(aR\) rescales the lowest spectral variation by \(a^{-2}\); therefore train on several radii and evaluate zero-shot at unseen radii. The cheapest experiment is a coordinate MLP fitting analytic radial fields or Poisson solutions on disks and balls. Record relative L2 error, boundary-condition violation, optimization steps, and zero-shot radius-transfer error. Success is lower error and faster convergence near the Dirichlet boundary without meaningful inference overhead.