Geometric Structures of Frequency Domains of Band-Limited Scaling Functions and Wavelets

The support of the Fourier transform of a wavelet is said to be its frequency domain. In the research of geometric structures of frequency domains of band-limited wavelets, it is well known that the frequency domain of any band-limited wavelet has a hole, in which the origin lies. In Zhang (J. Approx. Theory 148:128–147, 2007 ), we further study measures, densities, and diameters of frequency domains of band-limited wavelets. The measure of the frequency domain of any wavelet is ≥2 π . If the measure is 2 π , then such a wavelet is said to be a minimally supported frequency (MSF) wavelet. In this paper, we will show that the frequency domain of any band-limited MRA wavelet contains that of some MSF wavelet. Meanwhile, we will discuss the geometric structure of the frequency domain of the corresponding scaling function.