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arXiv · 2610.32044

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

Aussie Greene
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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 $\mathcal{O}(h^3)$ 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.

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Aussie Greene. 2026-10-23. Optimal Convergence of Quadratic Not-a-Knot Spline Interpolation on Arbitrary Partitions. https://arxiv.org/abs/2610.32044

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