Dimension functions of rationally dilated wavelets

Abstract

In A characterization of dimension functions of wavelets, Bownik, Rzeszotnik, and Speegle define and characterize the dimension function of an integer dilated wavelet in L²([special characters omitted]). Furthermore, in The wavelet dimension function for real dilations and dilations admitting non-MSF wavelets, Bownik and Speegle define the dimension function for a rationally (in fact, real) dilated wavelet in L²([special characters omitted]). In this dissertation, we extend these two previous works to their logical intersection by defining the dimension function of a rationally dilated wavelet in L²([special characters omitted]). We show that the definition previously given for integer dilated wavelets extends to include rationally dilated wavelets as well. We also show how the necessary (and sufficient) conditions for the dimension function of an integer dilated wavelet should be written to obtain necessary conditions for the dimension function of a rationally dilated wavelet.

Furthermore, we produce and characterize all wavelet sets in [special characters omitted] consisting of exactly two or three intervals, thus extending an example previously given by Speegle in Bownik's work On a problem of Daubechies.