Continuous Piecewise Linear Functions

Abstract

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on the m-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhere X = (X 1 , X 2 , . . . , X m ) is some random vector, {E j } j ∈J is a finite family of finite sets.

Riesz estimators

“…This paper depends substantially on our complimentary work Aliprantis et al (2002), where we develop a general theory of the Riesz space of continuous piecewise linear functions.…”

“…The mathematics behind the theory of Riesz estimators are those of Riesz spaces and Banach lattices. For details we refer the reader to ARTICLE IN PRESS our complimentary paper Aliprantis et al (2002) and the monographs Abramovich and Aliprantis (2002), Aliprantis and Border (1999), Aliprantis and Burkinshaw (2003), Schaefer (1974), Luxemburg and Zaanen (1971).…”

“…In the univariate case we have the following characterization of Riesz estimators; see Aliprantis et al (2002). …”

“…We call such functions (continuous) piecewise linear. The geometry of such functions are extensively studied in Aliprantis et al (2002). Remarkably, a simple consequence of the work in Aliprantis et al (2002) shows the following.…”

“…We summarize the work in Aliprantis et al (2002) and provide an important equivalent formulation (due to Ovchinnikov, 2002) of our statistical model. Let f 1 ; f 2 ; .…”