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arXiv · 2608.29781

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

Shih-Hao Huang·Jephian C. -H. Lin·ShengLi Tzeng·Tzu-Lun Yuan
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Abstract

We study the eigen-structure of the penalty matrix arising from cubic smoothing splines on equally spaced knots. Using purely matrix-theoretic arguments, we show that its positive eigenvalues are simple, that the associated eigenvectors alternate between even and odd, and that the eigenvector for the $k$th largest eigenvalue has exactly $k+1$ sign changes. The approach provides a direct and transparent alternative to existing variational proofs of the oscillation property. These results show that equally spaced knots support a spline basis with both a parity structure and an oscillation pattern.

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Shih-Hao Huang, Jephian C. -H. Lin, ShengLi Tzeng, Tzu-Lun Yuan. 2026-08-30. Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis. https://arxiv.org/abs/2608.29781

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