Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis

What the math gives to ML

The paper identifies a structured eigensystem for the cubic-spline roughness matrix on equally spaced one-dimensional knots: after removing constant and linear trends, the positive eigenvalues are simple, eigenvectors have alternating parity, and their oscillation count increases with eigenvalue order. This can be transferred into a fixed spectral basis for sequence or one-dimensional signal networks, where coefficients associated with highly oscillatory modes are explicitly controlled rather than learned in an unstructured coordinate basis. The most practical first use is a drop-in spectral regularizer or truncated parameterization for positional features, SSM inputs, or narrow sequence models; the basis can be precomputed once for each sequence length.