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


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.


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

Mathematics Subject Classification

42C40 65T60 


  1. 1.
    Bergmann, R.: The fast Fourier transform and fast wavelet transform for patterns on the torus. Appl. Comput. Harmon. Anal. 35, 39–51 (2013a). doi: 10.1016/j.acha.2012.07.007 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bergmann, R.: The multivariate periodic anisotropic wavelet library. (2013b)
  3. 3.
    R. Bergmann, Translationsinvariante Räume multivariater anisotroper Funktionen auf dem Torus, Dissertation, Universität zu Lübeck (2013c)Google Scholar
  4. 4.
    Bergmann, R., Prestin, J.: Multivariate anisotropic interpolation on the torus. In: Fasshauer, G.E., Schumaker, L.L. (Eds.) Approximation Theory XIV: San Antonio 2013, of Springer Proceedings in Mathematics & Statistics, vol. 83, pp. 27–44. Springer International Publishing, Cham (2014). doi: 10.1007/978-3-319-06404-8_3
  5. 5.
    de Boor, C., Höllig, K., Riemenschneider, S.: Box Splines. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dahlke, S., Steidl, G., Teschke, G.: The continuous shearlet transform in arbitrary space dimensions. J. Fourier Anal. Appl. 16, 340–364 (2010). doi: 10.1007/s00041-009-9107-8 CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Goh, S., Lee, S., Teo, K.: Multidimensional periodic multiwavelets. J. Approx. Theory 98, 72–103 (1999). doi: 10.1006/jath.1998.3279 CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Guo, K., Labate, D.: Analysis and detection of surface discontinuities using the 3D continuous shearlet transform. Appl. Comput. Harmon. Anal. 30, 231–242 (2010). doi: 10.1016/j.acha.2010.08.004 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Koh, Y.W., Lee, S.L., Tan, H.H.: Periodic orthogonal splines and wavelets. Appl. Comput. Harmon. Anal. 2, 201–218 (1995). doi: 10.1006/acha.1995.1014 CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Langemann, D., Prestin, J.: Multivariate periodic wavelet analysis. Appl. Comput. Harmon. Anal. 28, 46–66 (2010). doi: 10.1016/j.acha.2009.07.001 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Maximenko, I.E., Skopina, M.A.: Multivariate periodic wavelets. St. Petersb. Math. J. 15, 165–190 (2003). doi: 10.1090/S1061-0022-04-00808-8 CrossRefGoogle Scholar
  12. 12.
    Mhaskar, H.N., Prestin, J.: On the detection of singularities of a periodic function. Adv. Comput. Math. 12, 95–131 (2000). doi: 10.1023/A:1018921319865 CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Narcowich, F.J., Ward, J.D.: Wavelets associated with periodic basis functions. Appl. Comput. Harmon. Anal. 3, 40–56 (1996). doi: 10.1006/acha.1996.0003 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Novikov, I.Y., Skopina, M.A., Protasov, V.Y.: Wavelet Theory, Translations of Mathematical Monographs 239. American Mathematical Society (2011)Google Scholar
  15. 15.
    Plonka, G., Tasche, M.: Periodic wavelets, Preprint 93/11 der Preprintreihe des FB Mathematik, Universität Rostock (1993)Google Scholar
  16. 16.
    Plonka, G., Tasche, M.: A unified approach to periodic wavelets. In: Chui, C.K., Montefusco, L., Puccio, L. (eds.) Wavelets: Theory, Algorithms and Applications. Wavelet Analysis and Its Applications, vol. 5, pp. 137–151. Academic Press, New York (1994)CrossRefGoogle Scholar
  17. 17.
    Prestin, J., Selig, K.: Interpolatory and orthonormal trigonometric wavelets. In: Zeevi, Y., Coifman, R. (eds.) Signal and Image Representation in Combined Spaces. Wavelet Analysis and Its Applications, vol. 7, pp. 201–255. Academic Press, New York (1998). doi: 10.1016/S1874-608X(98)80009-5 CrossRefGoogle Scholar
  18. 18.
    K. Selig, Periodische Wavelet-Packets und eine gradoptimale Schauderbasis, Dissertation, Universität Rostock (1998)Google Scholar

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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