Two Dimensional Trigonomentric Wavelets For Hermite Interpolation

The study of trigonomentric series(Fourier series) has taken an important part in the theory of advanced mathematics. On the other hand, wavelet analysis is valued more and more in the phylogeny of applied mathematics. But there is a nodus that how to conbine trigonomentric series with wavelet analysis, namely, how to constrct the trigonomentric wavelet. In 1995, one dimensional trigonomentric wavelet space had been researched and its properties had been derivatived in detail in the published study of Trigonometric Wavelets For Her-mite Interpolation by Ewald Quak. However, today one dimensional trigonomentric wavelet space haven't satisfied with the need for the theory and practice. The study of higher dimensional trigonomentric wavelet space has attracted a number of authors. This thesis is a explicit description for two dimensional trigonomentric wavelet for Hermite interpolation-space. This study consists of four chapters, and the main contents of each chapter are as follows:In the prolegomena, we outline the background of wavelet analysis and current research situations, difficulties of wavelet theory, as well as the work which will be completed in this study.In Chapter 1, we discuss the basic concept of wavelet and the variational time-frequency window of wavelet transform , introducing the others of wavelet,such as MRA, the decomposition and reconstrction of wavelet,biorthogonal wavelet.In Chapter 2, two dimensional separable wavelet space is researched. Firstly, from the concept of tensor product we constrct two dimensional MRA and the basic functions of two dimensional scalling space and wavelet space.Then we get the decomposition and reconstrction for binary function.In Chapter 3, firstly, we introduce Fourier series and its convergence property. Then from two dimensional Dirichlet kernel, we constrct two dimensional scalling function and wavelet function for Hermite interpolation and define the two dimensional Hermite interpolation operator. Finally, we get the Hermite interpolation properties of scalling function and wavelet function, which are used to formulate the coefficients of the dual scalling function and wavelet scalling function.