A Characterization of Nonuniform Multiwavelets Using Dimension Function


In this article, we present a characterization of nonuniform multiwavelets associated to a nonuniform multiresolution analysis (NUMRA) having finite multiplicity in terms of its dimension function. This, in turn, improves the main result of Gabardo and Yu given in (J Math Anal Appl 323(2):798–817, 2006). The concept of NUMRA was introduced by Gabardo and Nashed in which the translation set is a spectrum that is no longer a group.

This is a preview of subscription content, access via your institution.


  1. 1.

    Auscher, P.: Solution of two problems on wavelets. J. Geom. Anal. 5(2), 181–236 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Calogero, A., Garrigós, G.: A characterization of wavelet families arising from biorthogonal MRA’s of multiplicity \(d\). J. Geom. Anal. 11(2), 187–217 (2001)

  3. 3.

    Gabardo, J.-P., Nashed, M.Z.: Nonuniform multiresolution analyses and spectral pairs. J. Funct. Anal. 158(1), 209–241 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Gabardo, J.-P., Nashed, M.Z.: An analogue of Cohen’s condition for nonuniform multiresolution analyses, wavelets, multiwavelets, and their applications (San Diego, CA, 1997), pp. 41–61. Contemp. Math. 216, Amer. Math. Soc., Providence (1998)

  5. 5.

    Gabardo, J.-P., Yu, X.: Wavelets associated with nonuniform multiresolution analyses and one-dimensional spectral pairs. J. Math. Anal. Appl. 323(2), 798–817 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Gripenberg, G.: A necessary and sufficient condition for the existence of a father wavelet. Studia Math. 114(3), 207–226 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Hernández, E., Weiss, G.: A First Course on Wavelets. Studies in Advanced Mathematics, CRC Press, Boca Raton (1996)

    Book  MATH  Google Scholar 

  8. 8.

    Mittal, S., Shukla, N.K.: Generalized nonuniform multiresolution analyses. Colloq. Math. (to be appear)

  9. 9.

    Mittal, S., Shukla, N.K., Atuloba, N.A.S.: Nonuniform multiresolution analyses with finite multiplicity (preprint)

  10. 10.

    Mittal, S., Shukla, N.K., Atlouba, N.A.S.: Nonuniform multiwavelet sets and multiscaling sets. Numer. Funct. Anal. Optim. 37(2), 253–276 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Shukla, N.K., Mittal, S.: Wavelets on the spectrum. Numer. Funct. Anal. Optim. 35(4), 461–486 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Wang, X.: The study of wavelets from the properties of their Fourier transforms. Thesis (Ph.D.)-Washington University in St. Louis (1995)

  13. 13.

    Yu, X.: Wavelet sets, integral self-affine tiles and nonuniform multiresolution analyses. Thesis (Ph.D.)–McMaster University (Canada) (2005)

Download references

Author information



Corresponding author

Correspondence to Shiva Mittal.

Additional information

The first author is supported by Libyan government for pursuing her Ph.D. thesis at SHIATS, Deemed to be University, Allahabad, India.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Atlouba, N.A.S., Mittal, S. & Shukla, N.K. A Characterization of Nonuniform Multiwavelets Using Dimension Function. Results Math 72, 1239–1255 (2017). https://doi.org/10.1007/s00025-016-0648-2

Download citation


  • MRA with multiplicity D
  • spectral pairs
  • multiwavelets
  • multiscaling functions
  • dimension function

Mathematics Subject Classification

  • 42C40
  • 65T60