Skip to main content
Log in

Tight frames of multidimensional wavelets

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations

$$\varphi ^s (x) = \sum\limits_{k \in \mathbb{Z}^n } {h_k^s \sqrt {|\det A|} \varphi ^1 } (Ax - k) for s = 1, \ldots ,q,$$

where ϕ1 is a scaling function for this multiresolution and ϕ2, …, ϕq (q=|det A |) are wavelets. Orthogonality conditions for ϕ1, …, ϕq naturally impose constraints on the scaling coefficients\(\{ h_k^s \} _{k \in \mathbb{Z}^n }^{s = 1, \ldots ,q} \), which are then called the wavelet matrix. We show how to reconstruct functions satisfying the scaling equations above and show that ϕ2, …, ϕq always constitute a tight frame with constant 1. Furthermore, we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Cohen, A. (1990). Ondelettes, analyses multirésolutions et filtres en quadrature.Ann. Inst. H. Poincairé-Analyse non linéaire. 7, 439–459.

    MATH  Google Scholar 

  2. Cohen, A. and Daubechies, I. (1993). Non-separable bidimensional wavelet bases,Revista Mat. Iberoamericana, 51–137.

  3. Daubechies, I. (1992). Ten Lectures on Wavelets.CBMS-NSF Reg. Conf. Ser. Appl. Math., SIAM, Philadelphia, PA.

  4. Gröchenig, K. (1994). Orthogonality Criteria for Compactly Supported Scaling Functions.Appl. Computational Harmonic Anal.,1, 242–245.

    Article  MATH  Google Scholar 

  5. Gröchenig, K. and Haas, A. (1994). Self-Similar Lattice Tilings.J. Fourier Analysis and Appl.,1, 131–170.

    Article  MATH  Google Scholar 

  6. Gröchenig, K. and Madych, W.R. (1992). Multiresolution Analysis, Haar Bases, and Self-Similar Tilings of ℝn.IEEE Trans. Inform. Theory. 38, 556–568.

    Article  MathSciNet  Google Scholar 

  7. Heller, P.N., Resnikoff, K.L. and Wells, Jr., R.O. (1992).Wavelet Matrices and the Representation of Discrete Functions. Wavelets: A Tutorial in Theory and Applications. Chui, C.K., Ed., 15–50.

  8. Lagarias, J.C. and Wang, Y. (1996). Self-Affine Tiles in ℝn.Adv. Math. 121, 21–49.

    Article  MATH  MathSciNet  Google Scholar 

  9. Lagarias, J.C. and Wang, Y. (1996). Integral Self-Affine Tiles in ℝn. I. Standard and nonstandard digit sets.J. London Math. Soc. 54, 161–179.

    MATH  MathSciNet  Google Scholar 

  10. Lagarias, J.C. and Wang, Y. Integral Self-Affine Tiles in ℝn. II. Lattice tilings. preprint.

  11. Lagarias, J.C. and Wang, Y. (1996). Haar Bases forL 2(ℝn) and Algebraic Number Theory.J.Number Theory.57, 181–197.

    Article  MATH  MathSciNet  Google Scholar 

  12. Lagarias, J.C. and Wang, Y. (1995). Haar-Type Orthonormal Wavelet Bases in ℝ2.J. Fourier Analysis and Appl. 2, 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  13. Lagarias, J.C. and Wang, Y. Orthogonality Criteria for Compactly Supported Scaling Functions in ℝn. preprint.

  14. Lawton, W.M. (1990). Tight frames of compactly supported affine wavelets.J. Math. Phys. 31, 1898–1901.

    Article  MATH  MathSciNet  Google Scholar 

  15. Lawton, W.M. (1991). Necessary and sufficient conditions for constructing orthonormal wavelet bases.J. Math. Phys. 32, 57–61.

    Article  MATH  MathSciNet  Google Scholar 

  16. Madych, W.R. (1992).Some Elementary Properties of Multiresolution Analyses of L 2(ℝn). Wavelets: A Tutorial in Theory and Applications. Chui, C.K., Ed., 259–294.

  17. Meyer, Y. (1992). Wavelets and operators.Cambridge studies in advanced mathematics. 37.

  18. Strichartz, R.S. (1993). Wavelets and self-affine tilings.Constructive Approx. 9, 327–346.

    Article  MATH  MathSciNet  Google Scholar 

  19. Wojtaszczyk, P. (1996). A Mathematical Introduction to Wavelets.London Mathematical Society Student Texts.37.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bownik, M. Tight frames of multidimensional wavelets. The Journal of Fourier Analysis and Applications 3, 525–542 (1997). https://doi.org/10.1007/BF02648882

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02648882

Math Subject Classifications

Keywords and Phrases

Navigation