Discrete approximation by first-degree splines with free knots
Theoretical foundation for optimal spline fitting with free knots, relevant to approximation theory and numerical analysis.
The paper proves existence and characterizes knot positions for best L_p approximation of discrete functions by first-degree splines with free knots, enabling a subsequent algorithm for L_2-norm global best approximation.
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.