Abstract
My work is about generalization of Haar-like wavelets. Our constructions are based on the special properties of three classical systems: Rademacher-, Walsh-Paley- and Haar-systems. We used a simple and general method to construct orthogonal or biorthogonal systems starting from the Dirichlet-kernel of the systems. Discrete orthogonal polynomial systems can be constructing by the Christoffel-Darboux-formula. Paley has proved that the 2^n-th Dirichlet-kernel of Walsh-Paley-system can be written in a simple form. This property based on the fact, the product system of Rademacher system is the Walsh system. The same formula is the key at the construction of Haar-like systems. The Fourier-coefficients with respect to UDMD product system can be computed by an FFT-like fast algorithm and a similar methode can be used for the reconstruction. At the end of the work we introduce new rational interpolation operators for the upper and lower half plane using the Malmquist-Takenaka systems of these Hardy spaces. Combining this two interpolations we can give exact interpolation for a large class of rational functions.
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