Framelet Sets and Associated Scaling Sets

In time-frequency analysis, an increasing interest is to develop various tools to split a signal into a set of non-overlapping frequency regions without the influence of their adjacent regions. Although the framelet is an ideal tool for time-frequency analysis, most of the framelets only give an overlapping partition of the frequency domain. In order to obtain a non-overlapping partition of the frequency domain, framelet sets and associated scaling sets are introduced. In this study, we will investigate the relation between framelet (or scaling) sets and the frequency domain of framelets (or frame scaling functions). We find that the frequency domain of any frame scaling function always contains a scaling set and the frequency domain of any FMRA framelet always contains a framelet set. Moreover, we give a simple approach to construct various framelet/scaling sets from band-limited framelets and frame scaling functions.