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Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane

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Blaschke Products and Their Applications

Part of the book series: Fields Institute Communications ((FIC,volume 65))

Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. NuHAG, Faculty of Mathematics, University of Vienna, Alserbachstrasse 23, 1090, Wien, Austria

    Hans G. Feichtinger & Margit Pap

  2. University of Pécs, Ifjúság útja 6, 7634, Pécs, Hungary

    Margit Pap

Authors
  1. Hans G. Feichtinger
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  2. Margit Pap
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Correspondence to Margit Pap .

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Editors and Affiliations

  1. , Département de mathématiques et de stati, Université Laval, avenue de la Médecine 1045, Quebec, G1K 7P4, Québec, Canada

    Javad Mashreghi

  2. , Institut Camille Jordan, Université Claude Bernard Lyon 1, avenue du 11 novembre 1918 43, Villeurbanne cedex, 69622, France

    Emmanuel Fricain

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Feichtinger, H.G., Pap, M. (2013). Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane. In: Mashreghi, J., Fricain, E. (eds) Blaschke Products and Their Applications. Fields Institute Communications, vol 65. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5341-3_11

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