Parseval frame scaling sets and MSF Parseval frame wavelets

In this paper, we consider the Parseval frame (PF) scaling sets and the MSF Parseval frame wavelets (PFWs) in L2ðRdÞ with dilations induced by expanding matrices A with integer coefficients of arbitrary determinant such that jdetAj ¼ 2. We firstly characterize the PF scaling sets, and then provide a method of construction of PF scaling sets. We also prove that all PF scaling sets arise in that way. Finally, by studying the relation between the MSF PFWs and the PF scaling sets, we derive that each PF scaling set S gives rise to a MSF PFW w, and furthermore each MSF PFW whose dimension function is essentially bounded by 1 arises from a PF scaling set and the corresponding PF MRA. Using our results, one can easily construct various PF scaling sets and MSF PFWs.