Spectral eigenvalue set of self-similar measures associated with product-form Hadamard triples

What the math gives to ML

The paper constructs Fourier spectra for self-similar measures generated by product-form Hadamard triples and characterizes scalars that preserve spectrality. The transferable asset is an exactly orthogonal, multiscale Fourier dictionary for a nonuniform fractal-like input distribution, together with multiplier sets that permit structured frequency rescaling without destroying orthogonality. This suggests replacing randomly sampled Fourier features or ad hoc positional encodings with a deterministic digit-tree encoding matched to the empirical coordinate distribution. The strongest initial target is neural fields or coordinate MLPs, where feature conditioning and multiscale coverage directly affect optimization and high-frequency reconstruction.