Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane
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A multiresolution analysis in the Hardy space of the unit disc was introduced recently (see Pap in J. Fourier Anal. Appl. 17(5):755–776, 2011). In this paper we will introduce an analogous construction in the Hardy space of the upper half plane. The levels of the multiresolution are generated by localized Cauchy kernels on a special hyperbolic lattice in the upper half plane. This multiresolution has the following new aspects: the lattice which generates the multiresolution is connected to the Blaschke group, the Cayley transform and the hyperbolic metric. The second: the nth level of the multiresolution has finite dimension (in classical affine multiresolution this is not the case) and still we have the density property, i.e. the closure in norm of the reunion of the multiresolution levels is equal to the Hardy space of the upper half plane. The projection operator to the nth resolution level is a rational interpolation operator on a finite subset of the lattice points. If we can measure the values of the function on the points of the lattice the discrete wavelet coefficients can be computed exactly. This makes our multiresolution approximation very useful from the point of view of the computational aspects.
KeywordsHyperbolic wavelets Multiresolution on the upper half plane Interpolation operator Orthogonal rational wavelets
Mathematics Subject Classification30H10 33C47 41A20 42C40 43A32 43A65
This chapter was developed during a fruitful stay of the second author at NuHAG group at the University of Vienna as Marie Curie fellow FP7-People-IEF-2009.
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