Characterizations Of The Weyl-Heisenberg Frame Sets

A Weyl-Heisenberg frame for L2(R) is a frame consisting of the set of translates and modulates of a fixed function in L2(R), i.e.,{EmbTnag}m,n∈z with a, b>0 and g∈L2(R). A special type of Weyl-Heisenberg frames in L2(R) consists of those whose mother func-tions are defined by set theoretic functions. The corresponding sets involved are called Weyl-Heisenberg frame sets. In this dissertation, we are concerned with the existence of Weyl-Heisenberg frame set. The characterizations on some special set which is a WH-frame set are obtained. Including the first chapter of introduction, there are five chapters in the thesis.In chapter 2, in the critical point, a=b=1, we use the Zak transform to judge whether a set is a WH-frame set, especially, when the set is an integer translations of unit interval or a self-affine tile.In chapter 3, we study the condition of the union of two intervals which is a WH-frame set in the critical point. By the Zak transform and the distribution of real, we give the complete characterization of this set which is a WH-frame set.In chapter 4, we investigate the difficult problem at the rationally oversampled case. Let E=[0,1)+{n1,..., nk} where{n1,..., nk}(?)Z. We show that E is a Weyl-Heisenberg frame set for (p/q,1) is equivalent to classifying the integer sets{n1,...,nk} such that p(z)=∑j=1k znj does not have any zeros on the unit circle in the plane. To show our results the technique in the Zak transform has been used.In the last chapter, we study the abc-problem in a Weyl-Heisenberg frame and obtain some new results when a is a rational number.