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Journal of Fourier Analysis and Applications

, Volume 21, Issue 2, pp 342–369 | Cite as

Multivariate Periodic Wavelets of de la Vallée Poussin Type

  • Ronny Bergmann
  • Jürgen Prestin
Article

Abstract

In this paper we present a general approach to multivariate periodic wavelets generated by scaling functions of de la Vallée Poussin type. These scaling functions and their corresponding wavelets are determined by their Fourier coefficients, which are sample values of a function, that can be chosen arbitrarily smooth, even with different smoothness in each direction. This construction generalizes the one-dimensional de la Vallée Poussin means to the multivariate case and enables the construction of wavelet systems, where the set of dilation matrices for the two-scale relation of two spaces of the multiresolution analysis may contain shear and rotation matrices. It further enables the functions contained in each of the function spaces from the corresponding series of scaling spaces to have a certain direction or set of directions as their focus, which is illustrated by detecting jumps of certain directional derivatives of higher order.

Keywords

Wavelets Lattices de la Vallée Poussin means Periodic multiresolution analysis Shift-invariant space 

Mathematics Subject Classification

42C40 65T60 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Technology KaiserslauternKaiserslauternGermany
  2. 2.Institute of MathematicsUniversity of LübeckLübeckGermany

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