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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 form

where X=(X1, X2, …, Xm) is some random vector, {Ej}jJ is a finite family of finite sets.

Corresponding author
Address correspondence to Charalambos D. Aliprantis, Department of Economics, Purdue University, West Lafayette, IN 47907-1310, USA; e-mail:
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Y.A. Abramovich and C.D. Aliprantis 2002a An invitation to operator theory, vol. 50 of Graduate Studies in Mathematics. Providence, RI: American Mathematical Society.
Y.A. Abramovich and C.D. Aliprantis 2002b Problems in operator theory, vol. 51 of Graduate Studies in Mathematics. Providence, RI: American Mathematical Society.
C.D. Aliprantis and K.C. Border 1999 Infinite-dimensional analysis: a hitchhiker's guide. Berlin: Springer-Verlag.
C.D. Aliprantis and O. Burkinshaw 2003 Locally solid Riesz spaces with applications to economics, vol. 105 of Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society.
C.D. Aliprantis, D. Harris, and R. Tourky 2006 Riesz estimators. Journal of Econometrics, Forthcoming.
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D.J. Brown, C.B. Huijsmans, and B. de Pagter 1991 Approximating derivative securities in f-algebras, in C.D. Aliprantis, K.C. Border, and W.A.J. Luxemburg, eds., Positive operators, Riesz Spaces, and Economics, Studies in Economic Theory, Vol. 2, pp. 171–177. Springer-Verlag, New York and Heidelberg.
W.A.J. Luxemburg and A. C. Zaanen 1971 Riesz spaces, Vol. I. Amsterdam: North-Holland Publishing.
S. Ovchinnikov 2002 Max-min representation of piecewise linear functions. Beirtrage zur Algebra und Geometrie 43 (1), 297302.
H.H. Schaefer 1974 Banach lattices and positive operators, vol. 215 of Die Grundlehren der mathematischen Wissenschaften, New York: Springer-Verlag. Band 215.
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Macroeconomic Dynamics
  • ISSN: 1365-1005
  • EISSN: 1469-8056
  • URL: /core/journals/macroeconomic-dynamics
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