Construction of Wavelets and Applications

A sequence of increasing translation invariant subspaces can be defined by the Haar-system (or generally by wavelets). The orthogonal projection to the subs- paces generates a decomposition (multiresolution) of a signal. Regarding the rate of convergence and the number of operations, this kind of decomposition is much more favorable then the conventional Fourier expansion. In this paper, starting from Haar-like systems we will introduce a new type of multire- solution. The transition to higher levels in this case, instead of dilation will be realized by a two-fold map. Starting from a convenient scaling function and two-fold map, we will introduce a large class of Haar-like systems. Besides others, the original Haar sys- tem and Haar-like systems of trigonometric polynomials, and rational functions can be constructed in this way. We will show that the restriction of Haar-like systems to an appropriate set can be identified by the original Haar-system. Haar-like rational functions are used for the approximation of rational transfer func- tions which play an important role in signal processing [Bokor1 1998, Schipp01 2003, Bokor3 2003, Schipp 2002].