On Supports Of Fourier Transforms Of Two Classes Of Scaling Functions

Scaling functions are very important in the construction of wavelets, which have been attracting many waveletters' interest. G. G. Walter in 1994 introduced W-type scaling functions related to 2I, where I denotes an identity matrix. Zhihua Zhang in 2007 addressed supports of Fourier transforms of scaling functions and W-type scaling functions related to 2I, and obtained a characterization of a bounded measurable sets in R~d being the supports of some scaling function and some W-type scaling function, respectively. For a general d×d dilation matrix D, due to interaction between coordinates of the vector when D acts on a vector, the study of properties of scaling functions and W-type scaling functions is more difficult than that of the case D=2I. Therefore, the results on a general dilation matrix D are not as rich as the results on D=2I. This thesis is devoted to the supports of Fourier transforms of scaling functions and W-type scaling functions related to a general dilation matrix D.Given a positive integer d. Let D be a d×d dilation matrix, and let G be a bounded measurable set in R~d. The following results are obtained in this thesis:Theorem 3.1.4 There exists a scaling functionφrelated to D such that Supp((?))=G if and only ifTheorem 3.2.3 There exists a W-type scaling functionφrelated to D such that Supp((?))=G if and only if Our proofs of Theorem 3.1.4 and Theorem 3.2.3 are both constructive. In the last chapter of this thesis, some examples for Theorem 3.1.4 and Theorem 3.2.3 are provided.