The Harmonic Transform

This paper introduces the Harmonic Transform (HT), a mathematically rigorous alternative to Fourier-based methods for signal representation on bounded domains. By utilizing curvature-weighted polynomials and a recurrence-based optimization, the HT achieves zero-leakage projection and spectral containment for shell-admissible signals. This framework is specifically suited for physical and computational systems where boundary conditions and geometric structure dominate global periodicity.