An Overview of Wavelet Based Multiresolution Analyses

In this paper we give an overview of some wavelet based multiresolution analyses. First, we briefly discuss the continuous wavelet transform in its simplest form. Then we give the definition of multiresolution analysis and show how wavelets fit into it. We take a closer look at orthogonal, biorthogonal and semiorthogonal wavelets. The fast wavelet transform, wavelets on closed sets (boundary wavelets), multidimensional wavelets and wavelet packets are discussed. Several examples of wavelet families are given and compared. Finally, the essentials of two major applications are outlined: data compression and compression of linear operators. Keywords: wavelets, multiresolution analysis, compression AMS(MOS) classification: 42-02, 42C10 1 Introduction Wavelets and wavelet techniques have recently generated much interest, both in applied areas as well as in more theoretical ones. The class of wavelet techniques is not really precisely defined and it keeps changing. Hence, it is virtually i..