# Discrete approximation by first-degree splines with free knots

## TL;DR

This paper investigates the approximation of discrete functions using first-degree splines with free knots across various norms, establishing existence, knot positioning, and developing an algorithm for optimal approximation in the L2 norm.

## Contribution

It proves the existence of best approximations, characterizes knot positions, and introduces an algorithm for global best approximation in the L2 norm.

## Key findings

- Existence of best approximations proven
- Characterization of knot positions provided
- Algorithm for optimal L2 approximation developed

## Abstract

This paper deals with the approximation of discrete real-valued functions by first-degree splines (broken lines) with free knots for arbitrary $L_p$-norms ($1 \leq p \leq \infty)$. We prove the existence of best approximations und derive statements on the position of the (free) knots of a best approximation. Building on this, elsewhere we develop an algorithm to determine a (global) best approximation in the $L_2$-norm.