Stability properties for a compactly supported prescale function

We show that if phi is a continuous, minimally supported prescale function, then its translates are linearly independent on any set of positive measur e in the unit interval. This generalizes results of Y. Meyer and P. G. Lema rie. This result implies that a stability condition, introduced by Gundy and Kaz arian for the study of local convergence of spline wavelet expansions, is s atisfied for all expansions arising from multiresolution analyses generated by such prescale functions phi.