On some properties of smooth sums of ridge functions

The following problem is studied: If a finite sum of ridge functions defined on an open subset of $\mathbb R^n$ belongs to some smoothness class, can one represent this sum as a sum of ridge functions (with the same set of directions) each of which belongs to the same smoothness class as the whole sum? It is shown that when the sum contains $m$ terms and there are $m-1$ linearly independent directions among $m$ linearly dependent ones, such a representation exists.