Measures, densities and diameters of frequency bands of scaling functions and wavelets

We investigate the measures of frequency bands of scaling functions and wavelets, and solve an open problem. Furthermore, we give conditions for which the measures of frequency bands of scaling functions are less than and the measures of frequency bands of wavelets are less than 4π. We also discuss the densities around the origin and the diameters of frequency bands of scaling functions and wavelets, and show that there are essential differences between the single and multidimensional cases.