Continuous Wavelet Transform And Its Applications

Wavelet theory concludes continuous wavelet transform and discrete wavelet transform. Continuous wavelet transform is an important part of wavelet theory. It is important that we continue to study continuous wavelet transform. In thise book, we generalize continuous wavelet transform from L2 (R) to a few function spaces that are wider than L2 (R). As an application of continuous wavelet transform, we discuss the relation between some differential equations and the integral equations.1. We discuss continuous wavelet transform on L2(R); get the condition which makes the reconstruction formula produced by the integral kernel of the function which is different from the transform function true; discuss the relation between the reconstruction formula produced by the integral kernel of the transform function and that produced by the integral kernel of the function which is different from the transform function.2. We generalize continuous wavelet transform fromL2(R)to L2(R"), vector function space respectively; get the conditions that make the reconstruction formula produced by the integral kernel of the transform function and that produced by the integral kernel of the function which is different from the transform function true not only in the weak topology but also in the strong topology; discuss the relation between the two reconstruction formulas.3. We generalize continuous wavelet transform from L2(R) to abstract function space; get the condition that makes the reconstruction formula produced by the integral kernel of the transform function true not only in the weak topology but also in the strong topology; generally speaking, we don't know what conditions that transform abstract functions satisfy make the reconstruction formula exist; get the condition of transform function that makes the reconstruction formula exist and be simplest; find a method that we look for the transform function.4. As an application of continuous wavelet transform, we discuss the relations between some differemtial equaltions and the integral equations by using the continuous wavelet transform in L2(R), L2(R") , vector function space and abstract function space respectively; prove that they are equivalent not only in the weak topology but also in the strong topology.