# Divided Differences, Falling Factorials, and Discrete Splines: Another Look at Trend Filtering and Related Problems

arXiv: https://arxiv.org/abs/2003.03886 · HTML: https://mlsanity.com/paper/2003.03886/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2020.

## Built on by (most cited first, 6 of 6)

- [Multivariate Trend Filtering for Lattice Data](https://mlsanity.com/paper/2112.14758/) (2021)
- [The Voronoigram: Minimax Estimation of Bounded Variation Functions From Scattered Data](https://mlsanity.com/paper/2212.14514/) (2022)
- [A Cross Validation Framework for Signal Denoising with Applications to Trend Filtering, Dyadic CART and Beyond](https://mlsanity.com/paper/2201.02654/) (2022)
- [Spatially Adaptive Online Prediction of Piecewise Regular Functions](https://mlsanity.com/paper/2203.16587/) (2022)
- [Whittaker-Henderson smoother for long satellite image time series interpolation](https://mlsanity.com/paper/2604.00048/) (2026)
- [A new approach to locally adaptive polynomial regression](https://mlsanity.com/paper/2412.19802/) (2024)

## Abstract (arXiv)

This paper reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by (a suitable notion of) continuity in divided differences. As it happens, discrete splines bear connections to a wide array of developments in applied mathematics and statistics, from divided differences and Newton interpolation (dating back to over 300 years ago) to trend filtering (from the last 15 years). We survey these connections, and contribute some new perspectives and new results along the way.