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Abstract Constructions of GMRAs

  • Kathy D. Merrill
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

This chapter presents abstract constructions of GMRAs and wavelets using direct sums and direct limits. As a first example, we take up super-wavelets, which were developed in direct sum spaces to handle the application of multiplexing, the sending of multiple signals on a carrier at the same time. Direct limits, the second construction technique we discuss, were first used to build classical MRA’s and wavelets from filters, and later generalized to construct abstract GMRA’s from multiplicity functions. Finally we present a technique that uses direct sums to find a classifying set for all GMRA’s with finite multiplicity function and Haar measure class.

References

  1. 1.
    Baggett, L., Courter, J., Merrill, K.: The construction of wavelets from generalized conjugate mirror filters in \(L^2(\mathbb R^n)\). Appl. Comput. Harmon. Anal. 13, 201–223 (2002)Google Scholar
  2. 2.
    Baggett, L., Furst, V., Merrill, K., Packer, J.: Generalized filters, the low-pass condition, and connections to multiresolution analysis. J. Funct. Anal. 257, 2760–2779 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baggett, L., Larsen, N., Merrill, K., Packer, J., Raeburn, I.: Generalized multiresolution analyses with given multiplicity functions. J. Fourier Anal. Appl. 15, 616–633 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baggett, L., Furst, V., Merrill, K., Packer, J.: Classification of generalized multiresolution analyses. J. Funct. Anal. 258, 4210–4228 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baggett, L., Larsen, N., Packer, J., Raeburn, I., Ramsay, A.: Direct limits, multiresolution analyses, and wavelets. J. Funct. Anal. 258, 2714–2738 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Beals, R.: Operators in functions spaces which commute with multiplications. Duke Math. J. 35, 353–362 (1968)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bildea, S., Dutkay, D., Picioroaga, G.: MRA super-wavelets. New York J. Math. 11, 1–19 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Courter, J.: Construction of dilation d wavelets. Contemp. Math. 247, 183–206 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dutkay, D.: The wavelet Galerkin operator. J. Oper. Theory 51, 49–70 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dutkay, D.: Low-pass filters and representations of the Baumslag-Solitar group. Trans. Am. Math. Soc. 358, 5271–5291 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dutkay, D., Jorgensen, P.: Wavelets on fractals. Rev. Mat. Iberoamericana 22, 131–180 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dutkay, D. Jorgensen, P.: Oversampling generates super-wavelets. Proc. Am. Math. Soc. 135, 2219–2227 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dutkay, D., Jorgensen, P.: Fourier series on fractals: a parallel with wavelet theory. Contemp. Math. 464, 76–101 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Han, D., Larson, D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147, x+94 (2000)Google Scholar
  15. 15.
    Larsen, N., Raeburn, I.: From filters to wavelets via direct limits. Contemp. Math. 414, 35–40 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Kathy D. Merrill
    • 1
  1. 1.Department of MathematicsThe Colorado CollegeColorado SpringsUSA

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