The Construction Of Bivariate Periodic Wavelets

Wavelets analysis is a new branch of mathematics developed from 80's last century which has been widely used in numerical analysis, engineering technology and other fields. It has become a focus of many subjects.Compare to the univariate wavelet, the theory of multivariate wavelet is not consummate. It is hard to construct which restrict its applications. Now the wavelets applied are most tensor product wavelets. There are a few practical non-tensor product wavelets. Therefore, the construction of multivariate wavelets especially the non-tensor product wavelets has been a hot problem of study.The classical wavelets are mostly defined on the whole space. But, many practical problems (such as the numerical solution for differential and integral equations, signal (image) processing and other fields) encountered in applications are always on finite region which may lead extra computations and boundary error when we deal with these problems with the wavelets defined on the whole space. Therefore, it is important to construct the wavelets on the finite region which have some good properties (such as smoothness. symmetry, cardinal interpolatory). Various efforts have been made to construct the wavelets on finite region ([30, 31]), among which constructing periodic wavelets is a important approach.Periodic wavelets were studied in Meyer~[32] and Daubechies~[33] first by pe-riodizing the known wavelets. Subsequently, many researchers contributed to the establishment and development of periodic wavelet theory. Chui-Mhaskar~[34] constructed the trigonometric wavelets. The general theory of periodic wavelet was studied first by Plonka-Tashe, Koh-Lee-Tan and Narcowich-Ward. They de-fined the periodic multiresolution analysis first, and then, constructed different periodic scaling functions and wavelets. The scaling functions given in [35] are semi-orthogonal while the scaling functions obtained in [36] are orthogonal but not equivalent by translated. Narcowich-Ward'37' obtained the scaling functions which are invariable by translated. The decomposition and reconstruction algorithms involve only 2 terms for the last two methods. Plonka and Chen et al did a lot of important work in constructing periodic wavelets. (see[38, 39, 40-46]). In [41], Chen et. al. constructed a class of periodic orthogonal wavelets which are real-valued. The decomposition and reconstruction algorithms involve only 4 terms. Subsequently, Chen and Xiao et. al. constructed a class of biorthogonal periodic wavelets which are real-valued, symmetric, cardinal interpolatory and local. (see[44-46]). But, the construction of the dual periodic scaling functions and wavelets and therefore the decomposition and reconstruction algorithms further need to compute the inner product of the original scaling functions.For the multivariate setting, Liang-Jin-Chen'47' construed a class of orthogonal periodic wavelets from Box spline functions which are complex. The corresponding decomposition and reconstruction algorithms involve only 4 terms. Goh-Lee-Teot48' give a general method to construct multivariate periodic multi-wavelets. Chen-Li'49! constructed multivariate biorthogonal periodic wavelets.The main topic of this paper is the construction of periodic wavelets. This paper consists of four parts. The construction of periodic wavelets on tangle region is the subject of the first two parts.In part one, we give a method for constructing a class of bivariate Box spline periodic wavelets derived from Box spline functions, which are real-valued, symmetric and orthogonal. There involve only 4 terms in the corresponding decomposition and reconstruction algorithms which are very simple for practical computation and implementation. The procedure for constructing is as follows.Step 1. Construct the periodic Box spline functions by periodizing the Box spline functions and generate a periodic multiresolution analysis of Ll(TT) fur-ther. Where TT is the rectangle of edge length T. Ll{TT) represents the set of all periodic, square-integral functions defined on TT.Step 2. Construct a basis of the scaling subspace by trigonometric transforms. It should be pointed out that these basis functions are not orthogonal. This is different from the case of univariate (In univariate case, the basis functions obtained by sine and cosine transforms are orthogonal (see [41])). However, we find that these basis functions have the equal norm.Step 3. Construct the orthonomal basis functions (orthonomal periodic scaling functions) of the scaling subspace and get the two scaling relations. Step 4. Construct the orthonomal periodic wavelets by matrix extension. Furthermore, we studied the relation between the periodic wavelets and the Fourier series. Finally, we give some examples.In part two, we are interested in generalizing the construction methods in part one and [41]. It should be pointed out that our construction method is not a simple generalization of the counterpart in univariate. We choose a class of general compactly supported refinable functions as the original function to replace the Box spline functions. Under some proper conditions, we prove that these functions can generate a periodic multiresolution analysis by proper periodization method. Then, we construct a class of real-valued orthogonal periodic wavelets by the similar method in part one. In step 4, we give two methods to implement the matrix extension. One of them is from Houshold transforms and the other is from matrix extension directly. The first method needs to compute the Houshold matrices while the second one needs no computation and therefore simplifies the complexity of computing.All of the periodic wavelets mentioned above are defined on product periodic regions. There are few wavelets defined on non-product periodic regions. However, many practical problems are based on non-product regions. Therefore, it is important to construct wavelets defined on non-product regions. In [50], Sun established a generalized Fourier analysis method on 3-directional hexagonal par-...