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Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis
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 kth 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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BibTeX
@misc{huang2026sturmliouvilletype,
title = {{Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis}},
author = {Shih-Hao Huang and Jephian C. -H. Lin and ShengLi Tzeng and Tzu-Lun Yuan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29781},
eprint = {2608.29781},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29781},
}
RIS
TY - UNPB
TI - Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis
AU - Shih-Hao Huang
AU - Jephian C. -H. Lin
AU - ShengLi Tzeng
AU - Tzu-Lun Yuan
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29781
UR - https://arxiv.org/abs/2608.29781
AB - 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 kth 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.
ER -