On nonlinear functions of linear combinations

Projection pursuit algorithms approximate a function of $p$ variables by a sum of nonlinear functions of linear combinations: \[ (1)\qquad f\left( {x_1 , \cdots ,x_p } \right) \doteq \sum_{i = 1}^n {g_i \left( {a_{i1} x_1 + \cdots + a_{ip} x_p } \right)} . \] We develop some approximation theory, give a necessary and sufficient condition for equality in (1), and discuss nonuniqueness of the representation.