Glasnik Matematicki, Vol. 46, No.1 (2011), 189-213.

SCALING SETS AND ORTHONORMAL WAVELETS WITH DILATIONS INDUCED BY EXPANDING MATRICES

Damir Bakić and Edward N. Wilson

Department of Mathematics, University of Zagreb, Zagreb, Croatia
e-mail: bakic@math.hr

Washington University in St. Louis, St. Louis, USA

Abstract.   The paper studies orthonormal wavelets in L2(Rn) with dilations induced by expanding integer matrices of arbitrary determinant. We provide a method for construction of all scaling sets and, hence, of all orthonormal MSF wavelets with the additional property that the core space of the underlying multiresolution structure is singly generated. Several examples on the real line and in R2 are included. We also prove that all MSF orthonormal wavelets whose dimension function is essentially bounded by 1 are obtained by our construction method. Finally, we derive a description of all wavelets (not necessarily MSF ones) that arise from a single scaling function in terms of the underlying multiresolution structure.

2000 Mathematics Subject Classification.   42A99, 42C15.

Key words and phrases.   Expanding matrix, orthonormal wavelet, scaling set, multiresolution analysis.

Full text (PDF) (free access)

DOI: 10.3336/gm.46.1.15

References:

  1. L. Baggett, H. Medina and K. Merrill, Generalized multiresolution analyses, and a construction procedure for all wavelet sets in Rn, J. Fourier Anal. Appl. 5 (1999), 563-573.
    MathSciNet     CrossRef

  2. D. Bakić, Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses, Appl. Comput. Harmon. Anal. 21 (2006), 281-304.
    MathSciNet     CrossRef

  3. D. Bakić, I. Krishtal and E .N. Wilson, Parseval frame wavelets with En(2)-dilations, Appl. Comput. Harmon. Anal. 19 (2005), 386-431.
    MathSciNet     CrossRef

  4. J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), 389-427.
    MathSciNet     CrossRef

  5. M. Bownik, The structure of shift-invariant subspaces of L2(Rn), J. Funct. Anal. 112 (2000), 282-309.
    MathSciNet     CrossRef

  6. M. Bownik and Z. Rzeszotnik, On the existence of multiresolution analysis for framelets, preprint, 2004.

  7. M. Bownik, Z. Rzeszotnik and D. Speegle, A characterization of dimension function of orthonormal wavelets, Appl. Comput. Harmon. Anal. 10 (2001), 71-92.
    MathSciNet     CrossRef

  8. X. Dai, Y. Diao, Q. Gu and D. Han, Wavelets with frame multiresolution analysis, J. Fourier Anal. Appl. 9 (2003), 39-48.
    MathSciNet     CrossRef

  9. X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998).
    MathSciNet    

  10. X. Dai, D. Larson and D. Speegle, Wavelet sets in Rn, J. Fourier Anal. Appl. 3 (1997), 451-456.
    MathSciNet     CrossRef

  11. X. Dai, D. Larson and D. Speegle, Wavelet sets in Rn, II, Contemp. Math. 216 (1998), 15-40.
    MathSciNet    

  12. Q. Gu and D. Han, On multiresolution analysis wavelets in Rn, J. Fourier Anal. Appl. 6 (2000), 437-447.
    MathSciNet     CrossRef

  13. E. Hernández and G. Weiss, A first course on wavelets, CRC Press, Boca Raton, 1996.
    MathSciNet     CrossRef

  14. M. Papadakis, On the dimension function of orthonormal wavelets, Proc. Amer. Math. Soc. 128 (2000), 2043-2049.
    MathSciNet     CrossRef

  15. M. Papadakis, H. Šikić and G. Weiss, The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts, J. Fourier Anal. Appl. 5 (1999), 496-521.
    MathSciNet     CrossRef

  16. P. Soardi and D. Weiland, Single wavelets in n dimensions, J. Fourier Anal. Appl. 4 (1998), 299-315.
    MathSciNet     CrossRef
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