On Orthogonal Wavelets with the Oversampling Property

In this note, we consider orthogonal wavelets with the oversampling property. We prove that if an orthogonal scaling function with exponential decay has the oversampling property, then it has the sampling property (i.e., it takes values 1 at 0 and 0 at other integers); therefore, an orthogonal scaling function with compact support has the oversampling property if and only if it is the Haar function.