Skip to main content
SpringerLink
Log in

Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces

  • Chapter
Sampling, Wavelets, and Tomography

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

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.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Aldroubi, Portraits of framesProc. Amer. Math. Soc. 123(1995), pp. 1661–1668.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Aldroubi, Oblique and hierarchical multiwavelet basesApplied Comp. Harmonic Anal.4 (1997), pp. 231–263.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Aldroubi, personal communication.

    Google Scholar 

  4. A. Aldroubi and M. Papadakis, Characterization and parameterization of multiwavelet basesContemp. Mathem. 216(1998), pp. 97–116.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. W. Baggett and K. D. Merrill, Abstract harmonic analysis and wavelets inR n Cont. Math.247 (2000), pp. 17–28.

    Article  MathSciNet  Google Scholar 

  8. J. J. Benedetto and M. Frazier (eds.), Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 1994.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  12. M. Bownik, The structure of shift invariant subspaces of L2(Rn)J. Funct. Anal. 177 (2000), pp. 282–309.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Casazza, The art of frame theoryTaiwanese J. Math.4 (2000), pp. 129–201.

    MathSciNet  MATH  Google Scholar 

  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. J. Courter, Construction of dilation-d waveletsCont. Math. 247(1999), pp. 183–205.

    Article  MathSciNet  Google Scholar 

  16. X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal waveletsMemoirs Amer. Math. Soc. 134, no.640, 1998.

    MathSciNet  Google Scholar 

  17. I. Daubechies, Framelets:MRA-based constructions of wavelet frames, preprint2001.

    Google Scholar 

  18. P. A. FillmoreA User’s Guide to Operator AlgebrasJohn Wiley & Sons, Inc., New York, 1996.

    MATH  Google Scholar 

  19. D. Han and D. R. Larson, Frames, bases and group representationsMemoirs Amer. Math. Soc. 147 no. 697, 2000.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  21. E. Hernández and G. WeissA First Course on WaveletsCRC Press, Boca Raton, FL, 1996.

    Book  Google Scholar 

  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. H. O. Kim and J. K. Lim, On frame wavelets associated with frame multiresolution analysisAppl. Comp. Harm. Anal.10 (2001), pp. 61–70.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Li, A theory of generalized multiresolution structure and pseudoframes of translatesJ. Fourier Anal. Appl.7 (2001), pp. 23–40.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Papadakis, On the Dimension function of orthonormal waveletsProc. Amer. Math. Soc. 128(2000), pp. 2043–2046.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. A. Petukhov, Explicit construction of framelets, preprint, 2000.

    Google Scholar 

  32. A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of L2(Rd)Canad. J. Math.47 (1995), 1051–1094.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Ron and Z. Shen, Compactly supported tight affine spline frames inL 2 (R d ), Math . Comp.67 (1998), pp. 191–207.

    MathSciNet  MATH  Google Scholar 

  35. I. W. Seleznik, Smooth wavelet tight frames with zero momentsAppl.Harm. Anal.10 (2001), pp. 163–181.

    Article  Google Scholar 

  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. T. Stavropoulos and M. Papadakis, On the multiresolution analyses of abstract Hilbert spacesBull. Greek Math. Soc.40 (1998), pp. 79–92.

    MathSciNet  MATH  Google Scholar 

  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. E. Weber, Frames and single wavelets for unitary groupsCanadian J. Mathem.54, pp. 634–647, 2002.

    Article  MATH  Google Scholar 

  40. G. Weiss and E. N. Wilson, The mathematical theory of wavelets, Proc. NATO-ASI meeting inHarmonic Analysis: A CelebrationKluwer, Dordrecht, 2001.

    Google Scholar 

Download references

Authors
  1. Manos Papadakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    John J. Benedetto

  2. Department of Mathematical Sciences, DePaul University, 60614-3250, Chicago, IL, USA

    Ahmed I. Zayed

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Papadakis, M. (2004). Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces. In: Benedetto, J.J., Zayed, A.I. (eds) Sampling, Wavelets, and Tomography. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8212-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-8212-5_8

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6495-8

  • Online ISBN: 978-0-8176-8212-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation