Trends in wavelet applications

The study of wavelet analysis which was formally developed in the late 1980s has progressed very rapidly. There exists a vast literature on its applications to image processing and partial differential equations. However, Black-Scholes equation of pricing, Maxwell's equations, variational inequalities, and complex dynamic optimization problems related to real-world phenomena are some of the areas where applications of wavelet methods have not been fully explored. Wavelet packet analysis which includes wavelet analysis as a special case has wide scope for further research from both theoretical and applied viewpoints. As we know, wavelet methods are refinements of Fourier analysis, finite element, and boundary element methods. Compared to classical methods, wavelet methods yield better results. We briefly introduce applications of wavelets to image processing and partial differential equations. Important publications have been cited, in particular those of Cohen, Dahmen, DeVore, and Meyer, and some of their contributions have been discussed. The other important topics such as Black-Scholes equation of option pricing, Maxwell's equation, varia-tional inequalities, modeling real-world problems, and complex dynamic optimization problems have been discussed, and some open problems are mentioned. The main aim of this chapter is to present an overview of certain applications of wavelets which may provide motivation for further research. We have briefly mentioned the properties of wavelet packets and have given updated references for further study.