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].