Optimal Convergence of Quadratic Not-a-Knot Spline Interpolation on Arbitrary Partitions

Abstract

The not-a-knot condition provides a simple spline interpolation scheme that requires no derivative data at the endpoints. Whereas the classical not-a-knot scheme using odd-degree splines selects knots that coincide with the data points, this approach does not extend to the even-degree case. In this paper we establish the well-posedness of the not-a-knot interpolation problem for even-degree splines, where knots are selected at midpoints of data values. We then show that the quadratic case, introduced in [M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981], achieves 𝒪(h³) convergence for C³ functions on arbitrary partitions without mesh-ratio restrictions. The proof derives a linear system defining the interpolating spline, normalizes the system to bound the matrix norm of its inverse using a Neumann-series argument, and establishes global error bounds.

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BibTeX RIS

BibTeX

@misc{greene2026optimal,
  title = {{Optimal Convergence of Quadratic Not-a-Knot Spline Interpolation on Arbitrary Partitions}},
  author = {Aussie Greene},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32044},
  eprint = {2610.32044},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32044},
}

RIS

TY  - UNPB
TI  - Optimal Convergence of Quadratic Not-a-Knot Spline Interpolation on Arbitrary Partitions
AU  - Aussie Greene
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32044
UR  - https://arxiv.org/abs/2610.32044
AB  - The not-a-knot condition provides a simple spline interpolation scheme that requires no derivative data at the endpoints. Whereas the classical not-a-knot scheme using odd-degree splines selects knots that coincide with the data points, this approach does not extend to the even-degree case. In this paper we establish the well-posedness of the not-a-knot interpolation problem for even-degree splines, where knots are selected at midpoints of data values. We then show that the quadratic case, introduced in [M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981], achieves 𝒪(h^3) convergence for C^3 functions on arbitrary partitions without mesh-ratio restrictions. The proof derives a linear system defining the interpolating spline, normalizes the system to bound the matrix norm of its inverse using a Neumann-series argument, and establishes global error bounds.
ER  -

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