# Characterization theorem for best polynomial spline approximation with free knots

arXiv: https://arxiv.org/abs/1412.2323 · HTML: https://mlsanity.com/paper/1412.2323/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2014.

## Built on by (most cited first, 1 of 1)

- [KAN vs LSTM Performance in Time Series Forecasting](https://mlsanity.com/paper/2511.18613/) (2025)

## Abstract (arXiv)

In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. We start from identifying a special property of the knots. Then, using this property, we construct a characterization theorem for best free knots polynomial spline approximation, which is stronger than the existing characterisation results when only continuity is required.