Boundary-Weighted Fourier Inequalities for Convex Domains

What the math gives to ML

The paper introduces a geometry-derived Fourier weight, \(\omega_\Omega(x)=m(\Omega\cap(2x-\Omega))\), measuring how much a frequency remains inside a reflected overlap of the convex bandwidth domain. Its inequalities control Fourier coefficients after multiplying them by the boundary factor \(\omega_\Omega^{-d}\), providing a principled distinction between robust interior frequencies and fragile frequencies near a spectral boundary. A direct neural-network transfer is a boundary-aware spectral loss or Fourier-layer gain that emphasizes errors and representations near the edge of an allowed band, where discretization and aliasing artifacts often concentrate. The first test should use a convex box or ball bandwidth, compare this construction against unweighted FFT loss and standard radial spectral weighting, and measure accuracy together with high-frequency error and training stability.