Hyperbolic Wavelets and Multiresolution in H^{2}(\mathbb{T}) |
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Authors: | Margit Pap |
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Institution: | 1. Marie Curie fellow, NuHAG Faculty of Mathematic, University of Vienna, Alserbachstrae 23, 1090, Wien, Austria 2. University of P??cs, Ifj??s??g ??tja 6, 7634, P??cs, Hungary
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Abstract: | In signal processing and system identification for
H2(\BbbT)H^{2}(\Bbb{T}) and
H2(\BbbD)H^{2}(\Bbb{D}) the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal
bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are constructed
from rational Blaschke functions, which form a group with respect to function composition that is isomorphic to the Blaschke
group, respectively to the hyperbolic matrix group. Consequently, the background theory uses tools from non-commutative harmonic
analysis over groups and the generalization of Fourier transform uses concepts from the theory of the voice transform. The
successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may
cause a drawback of the method. In this paper we give a set of poles and using them we will generate a multiresolution in
H2(\BbbT)H^{2}(\Bbb{T}) and
H2(\BbbD)H^{2}(\Bbb{D}). The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice
transform generated by a representation of the Blaschke group over the space
H2(\BbbT)H^{2}(\Bbb{T}). The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of
signals belonging to
H2(\BbbT)H^{2}(\Bbb{T}) and
H2(\BbbD)H^{2}(\Bbb{D}) if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties
of the hyperbolic wavelet representation will be studied. |
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Keywords: | |
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