Discrete approximation by first-degree splines with free knots

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.