Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces

  • Manos Papadakis
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We define a very generic class of multiresolution analysis of abstract Hilbert spaces. Their core subspaces have a frame produced by the action of an abelian unitary group on a countable frame multiscaling vector set, which may be infinite. We characterize all the associated frame multiwavelet vector sets and we generalize the concept of low and high pass filters. We also prove a generalization of the quadratic (conjugate) mirror filter condition, and we give two algorithms for the construction of the high pass filters associated to a given low pass filter.


High Pass Filter Riesz Basis Dual Frame Frame Wavelet Range Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Aldroubi, Portraits of framesProc. Amer. Math. Soc. 123(1995), pp. 1661–1668.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Aldroubi, Oblique and hierarchical multiwavelet basesApplied Comp. Harmonic Anal.4 (1997), pp. 231–263.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Aldroubi, personal communication.Google Scholar
  4. [4]
    A. Aldroubi and M. Papadakis, Characterization and parameterization of multiwavelet basesContemp. Mathem. 216(1998), pp. 97–116.MathSciNetCrossRefGoogle Scholar
  5. [5]
    L. Baggett, A. Carey, W. Moran, and P. Ohring, General existence theorems for orthonormal wavelets, an abstract approachPubl. Inst. Math. Sci. Kyoto Univ. 31(1995), pp. 95–111.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    L. W. Baggett, H. A. Medina, and K. D. Merrill, Generalized multiresolution analyses and a construction procedure for all wavelet sets inW n J. Fourier Anal. Applic. 5 (1999), pp.563–573.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    L. W. Baggett and K. D. Merrill, Abstract harmonic analysis and wavelets inR n Cont. Math.247 (2000), pp. 17–28.MathSciNetCrossRefGoogle Scholar
  8. [8]
    J. J. Benedetto and M. Frazier (eds.), Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 1994.zbMATHGoogle Scholar
  9. [9]
    J.J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banksAppl. Comp. Harm. Anal.5 (1998), pp. 389–427.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. J. Benedetto and O. M. Treiber, Wavelet frames: Multiresolution analysis and the unitary extension principle, inWavelet Transforms and Time-Frequency Signal AnalysisL. Debnath (ed.), Birkhauser, Boston, MA, pp. 3–36, 2001.CrossRefGoogle Scholar
  11. [11]
    C. de Boor, R. DeVore, and A. Ron, Approximation from shift-invariant subspaces of L2(Rd)Trans. Amer. Math. Soc. 341 (1994), pp. 787–806.MathSciNetzbMATHGoogle Scholar
  12. [12]
    M. Bownik, The structure of shift invariant subspaces of L2(Rn)J. Funct. Anal. 177 (2000), pp. 282–309.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    P. Casazza, The art of frame theoryTaiwanese J. Math.4 (2000), pp. 129–201.MathSciNetzbMATHGoogle Scholar
  14. [14]
    C. Chui, W. He, J. Stöckler, and Q. Sun, Compactly supported tight affine frames with integer dilations and maximal vanishing moments, to appear in Advances Comput.Maihem.special issue on frames, 2002.Google Scholar
  15. [15]
    J. Courter, Construction of dilation-d waveletsCont. Math. 247(1999), pp. 183–205.MathSciNetCrossRefGoogle Scholar
  16. [16]
    X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal waveletsMemoirs Amer. Math. Soc. 134, no.640, 1998.MathSciNetGoogle Scholar
  17. [17]
    I. Daubechies, Framelets:MRA-based constructions of wavelet frames, preprint2001.Google Scholar
  18. [18]
    P. A. FillmoreA User’s Guide to Operator AlgebrasJohn Wiley & Sons, Inc., New York, 1996.zbMATHGoogle Scholar
  19. [19]
    D. Han and D. R. Larson, Frames, bases and group representationsMemoirs Amer. Math. Soc. 147 no. 697, 2000.MathSciNetGoogle Scholar
  20. [20]
    D. Han, D. R. Larson, M. Papadakis, and Th. Stavropoulos, Multiresolution analysis of abstract Hilbert spaces and wandering subspacesCont. Math. 247(1999), pp. 259–284.MathSciNetCrossRefGoogle Scholar
  21. [21]
    E. Hernández and G. WeissA First Course on WaveletsCRC Press, Boca Raton, FL, 1996.CrossRefGoogle Scholar
  22. [22]
    R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. II, GSM Vol.16American Mathematical Society, 1997.Google Scholar
  23. [23]
    H. O. Kim and J. K. Lim, On frame wavelets associated with frame multiresolution analysisAppl. Comp. Harm. Anal.10 (2001), pp. 61–70.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. Li, A theory of generalized multiresolution structure and pseudoframes of translatesJ. Fourier Anal. Appl.7 (2001), pp. 23–40.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    M. Paluszynski, H. Sikic, G. Weiss, and S. Xiao, Generalized low pass filters and MRA frame wavelets, 7.Geom. Anal. 11(2001), pp. 311–342.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Papadakis, On the Dimension function of orthonormal waveletsProc. Amer. Math. Soc. 128(2000), pp. 2043–2046.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Papadakis, Generalized frame multiresolution analysis of abstract Hilbert spaces and its applications, SPIE Proc. 4119Wavelet Applications in Signal and Image Processing VIIIA. Aldroubi, A. Laine, and M. Unser (eds.), pp. 165–175, 2000.Google Scholar
  28. [28]
    M. Papadakis, Frames of translates and the generalized frame multiresolution analysis, Trends in Approximation Theory, K. Kopotun, T. Lyche, and M. Neamtu (eds.), pp. 353–362, Vanderbilt University Press, Nashville, TN, 2001.Google Scholar
  29. [29]
    M. Papadakis, Frames of translates and examples in generalized frame multiresolution analysis, SPIE Proc. 4478Wavelet Applications in Signal and Image Processing IXA. Aldroubi, A. Laine, and M. Unser (eds.), pp. 142–150, 2001.Google Scholar
  30. [30]
    M. Papadakis, G. Gogoshin, I. Kakadiaris, D.J. Kouri, and D. K. Hoffman, Radial frame multiresolution analysis in multidimensions, preliminary version, 2002.Google Scholar
  31. [31]
    A. Petukhov, Explicit construction of framelets, preprint, 2000.Google Scholar
  32. [32]
    A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of L2(Rd)Canad. J. Math.47 (1995), 1051–1094.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Ron and Z. Shen, Affine systems inL 2 (R d the analysis of the analysis operatorJ. Funct. Anal. 148(1997), pp. 408–447.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. Ron and Z. Shen, Compactly supported tight affine spline frames inL 2 (R d ), Math . Comp.67 (1998), pp. 191–207.MathSciNetzbMATHGoogle Scholar
  35. [35]
    I. W. Seleznik, Smooth wavelet tight frames with zero momentsAppl.Harm. Anal.10 (2001), pp. 163–181.CrossRefGoogle Scholar
  36. [36]
    I. W. Seleznik and L. Sendur, SPIE Proc.4119(2000),Wavelet Applications in Signal and Image Processing, A. Aldroubi, A. Laine, and M. Unsef (eds.).Google Scholar
  37. [37]
    T. Stavropoulos and M. Papadakis, On the multiresolution analyses of abstract Hilbert spacesBull. Greek Math. Soc.40 (1998), pp. 79–92.MathSciNetzbMATHGoogle Scholar
  38. [38]
    X. Wang, The study of wavelets from the properties of their Fourier transforms, Ph.D. Thesis, Washington University in St. Louis, 1995.Google Scholar
  39. [39]
    E. Weber, Frames and single wavelets for unitary groupsCanadian J. Mathem.54, pp. 634–647, 2002.CrossRefzbMATHGoogle Scholar
  40. [40]
    G. Weiss and E. N. Wilson, The mathematical theory of wavelets, Proc. NATO-ASI meeting inHarmonic Analysis: A CelebrationKluwer, Dordrecht, 2001.Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Manos Papadakis

There are no affiliations available

Personalised recommendations