Wavelet sets and frame sets

In this dissertation, some basic properties of wavelet sets and frame sets have been investigated. All the new results contained in the dissertation have been obtained in the past year under the supervision and the guidance of my thesis advisor Dr. Larson. The research centers on some closely related topics in wavelet theory and frame theory. These topics are initiated mainly by Dr. Larson's test questions, AMS Memoirs of Dr. Dai and Dr. Larson, and Dr. Han and Dr. Larson, a paper of Dr. Dai, Dr. Larson and Dr. Speegle and a recent paper by Dr. Han. In the AMS Memoirs by Dr. Dai and Dr. Larson, the authors introduced the concept of wavelet sets, interpolation families and operator interpolation of wavelets along with abundant examples of wavelet sets and interpolation maps. In the AMS Memoirs by Dr. Han and Dr. Larson, the authors introduced the concept of frame sets, strong complementary pairs of frame sets among other concepts. In the paper by Dr. Dai, Dr. Larson and Dr. Speegle, the existence of wavelet sets in high dimensional spaces was proved using a constructive argument. In an attempt to investigate wavelet sets and frame sets from a viewpoint of maps on $R$, we develop the concept of pseudo-interpolation maps and give a necessary condition for a pseudo-interpolation map to be an interpolation map associated with a pair of frame sets. Along the way of the paper by Dr. Dai, Dr. Larson and Dr. Speegle and the recent paper by Dr. Han, we also obtain a way to construct wavelet sets, which is quite convenient for applications. With the help of this construction method, we construct counterexamples to some questions raised by Dr. Larson and to an open question in Dr. Han's recent paper concerning interpolation pairs of wavelet sets and strong complementary pairs of frame sets. A certain property of interpolation pairs of wavelet sets has been pointed out.