In this paper we consider the problem of least squares approximation of scattered data over triangulations. We define finite dimensional space $S_1^0(\triangle)$ of continuous piecewise linear functions over a triangulation $\triangle$ and equip it with a basis. The basis consists of functions with local supports and pyramid-shaped graphs. Data are approximated by a function $f \in S_1^0(\triangle)$, which is represented as a linear combination of basis functions. The coefficients of the function are determined using the least squares method. We derive that coefficients of a function $f$ can be computed with solving an overdetermined system. The overdetermined system can be solved using the corresponding normal system determined by a symmetric sparse matrix. Its analysis ensures the existence and uniqueness of the approximation function.
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