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The construction ofr-regular wavelets for arbitrary dilations

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Abstract

Given a dilation matrix A and a natural number r we construct an associated r-regular multiresolution analysis with r-regular wavelet basis. Here a dilation is an n×n expansive matrix A (all eigenvalues λ of A satisfy |λ|>1) with integer entries. This extends a theorem of Strichartz which assumes the existence of a self-affine tiling associated with the dilation A. We also prove that regular wavelets have vanishing moments.

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References

  1. Auscher, P. (1992). Toute base d'ondelettes régulières deL 2(ℝ) est issue d'une analyse multi-résolution régulière,C.R. Acad. Sci. Paris Sér. I Math.,315, 1227–1230.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  3. Ayache, A. (1999). Construction of non separable dyadic compactly supported orthonormal wavelet bases forL 2(ℝ2) of arbitrarily high regularity,Rev. Mat. Iberoamericana,15, 37–58.

    MATH  MathSciNet  Google Scholar 

  4. Belogay, E. and Wang, Y. (1999). Arbitrarily smooth orthogonal nonseparable wavelets in ℝ2,SIAM J. Math. Anal.,30, 678–697.

    Article  MATH  MathSciNet  Google Scholar 

  5. Battle, G. (1989). Phase space localization theorem for ondelettes,J. Math. Phys.,30, 2195–2196.

    Article  MATH  MathSciNet  Google Scholar 

  6. Bownik, M. (1997). Tight frames of multidimensional wavelets,J. Fourier Anal. Appl.,3, 525–542.

    MATH  MathSciNet  Google Scholar 

  7. Bownik, M. (2000). The structure of shift invariant subspaces ofL 2(ℝn),J. Funct. Anal.,177, 282–309.

    Article  MATH  MathSciNet  Google Scholar 

  8. Bownik, M. (2000). The anisotropic Hardy spaces and wavelets, Ph.D. thesis, Washington University.

  9. Bownik, M., Rzeszotnik, Z., and Speegle, D.M. (2001). A characterization of dimension functions of wavelets,Appl. Comput. Harmon. Anal.,10, 71–92.

    Article  MATH  MathSciNet  Google Scholar 

  10. Bownik, M. and Speegle, D.M. Meyer type wavelet bases in ℝ2, preprint.

  11. Cabrelli, C., Heil, C., and Molter, U. (1998). Accuracy of lattice translates of several multidimensional refinable functions.J. Approx. Theory,95, 5–52.

    Article  MATH  MathSciNet  Google Scholar 

  12. Cohen, A., Gröchenig, K., and Villemoes, L.F. (1999). Regularity of multivariate refinable functions,Constr. Approx.,15, 241–255.

    Article  MATH  MathSciNet  Google Scholar 

  13. Conze, J. -P., Hervé, L., and Raugi, A. (1997). Pavages auto-affines, opérateurs de transfert et critères de réseau dans ℝd,Bol. Soc. Brasil. Mat., (N.S.),28, 1–42.

    Article  MATH  MathSciNet  Google Scholar 

  14. Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets,Comm. Pure Appl. Math.,41, 909–996.

    MATH  MathSciNet  Google Scholar 

  15. Daubechies, I. (1992). Ten lectures on wavelets,Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

    MATH  Google Scholar 

  16. Gröchenig, K. and Haas, A. (1994). Self-similar lattice tilings,J. Fourier Anal. Appl.,1, 131–170.

    Article  MATH  MathSciNet  Google Scholar 

  17. 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  MATH  MathSciNet  Google Scholar 

  18. Hernández, E. and Weiss, G. (1996). A first course on wavelets,Studies in Advanced Mathematics, CRC Press LLC, Boca Raton, FL.

    MATH  Google Scholar 

  19. Kenyon R., Li, J., Strichartz, R.S., and Wang, Y. (1999). Geometry of self-affine tiles, II,Indiana Univ. Math. J.,48, 25–42.

    MATH  MathSciNet  Google Scholar 

  20. Lagarias, J. C. and Wang, Y. (1995). Haar type orthonormal wavelet bases in ℝ2,J. Fourier Anal. Appl.,2, 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  21. 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 

  22. Lagarias, J.C. and Wang, Y. (1999). Corrigendum and addendum to: Haar bases forL 2(ℝn) and algebraic number theory,J. Number Theory,76, 330–336.

    Article  MATH  MathSciNet  Google Scholar 

  23. Lagarias, J.C., and Wang, Y. (1996). Self-affine tiles in ℝn,Adv. Math.,121, 21–49.

    Article  MATH  MathSciNet  Google Scholar 

  24. 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 

  25. Lagarias, J.C. and Wang, Y. (1997). Integral self-affine tiles in ℝn, II, lattice tilings,J. Fourier Anal. Appl.,3, 83–102.

    Article  MATH  MathSciNet  Google Scholar 

  26. Lemarié-Rieusset, P. -G. (1994). Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions,Rev. Mat. Iberoamericana,10, 283–347.

    MATH  MathSciNet  Google Scholar 

  27. Marcus, M. (1973). Finite dimensional multilinear algebra, Part 1,Pure and Applied Mathematics, Vol. 23, Marcel-Dekker Inc., New York.

    MATH  Google Scholar 

  28. Meyer, Y. (1992).Wavelets and Operators, Cambridge University Press, Cambridge.

    Google Scholar 

  29. Meyer, Y. and Coifman, R. (1997).Wavelets., Calderón-Zygmund and Multilinear Operators, Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  30. Potopia, A. (1997).A Problem of Lagarias and Wang, Master's thesis at Siedlce University, (Polish).

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Strichartz, R.S. and Wang, Y. (1999). Geometry of self-affine tiles, I,Indiana Univ. Math. J.,48, 1–23.

    MATH  MathSciNet  Google Scholar 

  33. Szlenk, W. (1984). An introduction to the theory of smooth dynamical systems, translated from Polish by Marcin E. Kuczma,PWN—Polish Scientific Publishers, Warsaw.

    Google Scholar 

  34. Wojtaszczyk, P. (1997).A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 525 East University Ave., 48109-1109, Ann Arbor, MI

    Marcin Bownik

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  1. Marcin Bownik
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Communicated by Y. Wang

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Bownik, M. The construction ofr-regular wavelets for arbitrary dilations. The Journal of Fourier Analysis and Applications 7, 489–506 (2001). https://doi.org/10.1007/BF02511222

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  • DOI: https://doi.org/10.1007/BF02511222

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