O. Davydov, M. Sommer and H. Strauss, Interpolation by bivariate linear splines, J. Comput. Appl. Math. 119 (2000), 115-131.

Abstract: We give a characterization of Lagrange interpolation sets for the spaces of continuous bivariate linear splines on regular triangulations. The characterization is based on a complete description of the zero sets of such splines.

Preprint version available:    pdf