Generalized Filters

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


Filters in a GMRA encode the containments V−1 ⊂ V0 and W−1 = δ−1W0 ⊂ V0 as relationships between scaling functions, wavelet functions, and their dilates. Classical filters were defined in \(L^2(\mathbb R)\) in terms of Fourier transforms of these functions, and were used to build MRA’s and orthonormal wavelets with desirable properties. Generalized filters take advantage of the GMRA structure by using the unitary operator given by spectral multiplicity in place of the Fourier transform.


  1. 1.
    Aldroubi A., Cabrelli, C., Molter, U.: Wavelets on irregular grids with arbitrary dilation and frame atoms for \(L^2(\mathbb R^d)\). Appl. Comput. Harmon. Anal. 17, 119–140 (2004)Google Scholar
  2. 2.
    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
  3. 3.
    Baggett, L., Jorgensen, P., Merrill, K., Packer, J.: Construction of Parseval wavelets from redundant filter systems. J. Math. Phys. 46, 1–28 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baggett, L., Jorgensen, P., Merrill, K., Packer, J.: A non-MRA Cr frame wavelet with rapid decay. Acta Appl. Math. 89, 251–270 (2006)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    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
  7. 7.
    Benedetto, J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5, 389–427 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Benedetto, J., Treiber, O.: Wavelet frames: multiresolution analysis and extension principles. In: Wavelet Transforms and Time-Frequency Analysis, pp. 3–36. Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  9. 9.
    Bownik, M.: Riesz wavelets and generalized multiresolution analyses. Appl. Comput. Harmon. Anal. 14, 181–194 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bownik, M., Rzeszotnik, Z.: Construction and reconstruction of tight framelets and wavelets via matrix mask functions. J. Funct. Anal. 256, 1165–1105 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bownik, M., Speegle, D.: Meyer type wavelet bases in \(\mathbb R^2\). J. Approx. Theory 116, 49–75 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bratteli, O., Jorgensen, P.E.T.: Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N. Integr. Equ. Oper. Theory 28, 382–443 (1997)zbMATHGoogle Scholar
  13. 13.
    Bratteli, O., Jorgensen, P.E.T.: Wavelets Through a Looking Glass. Birkhäuser, Boston (2002)CrossRefGoogle Scholar
  14. 14.
    Calogero, A.: Wavelets on general lattices, associated with general expanding maps of \(\mathbb R^n\). Electron. Res. Announc. Am. Math. Soc. 5, 1–10 (1999)Google Scholar
  15. 15.
    Cohen, A.: Wavelets and Multiscale Signal Processing. Chapman and Hall, London (1995)CrossRefGoogle Scholar
  16. 16.
    Courter, J.: Construction of dilation d wavelets. Contemp. Math. 247, 183–206 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Esteban, D., Galand, C.: Application of quadrature mirror filters to split band voice coding systems. In: IEEE International Conference on ICASSP’77, Washington D.C., pp. 191–195 (1977)Google Scholar
  19. 19.
    Furst, V.: A characterization of semiorthogonal Parseval wavelets in abstract Hilbert spaces. J. Geom. Anal. 17, 569–591 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hernandez, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996)CrossRefGoogle Scholar
  21. 21.
    Hernández, E., Wang, X., Weiss, G.: Smoothing minimally supported frequency wavelets II. J. Fourier Anal. Appl. 3, 23–41 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lawton, W.M.: Tight frames of compactly supported affine wavelets. J. Math. Phys. 31, 1898–1901 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mallat, S.: Multiresolution approximations and wavelet orthonormal bases of \(L^2(\mathbb R)\). Trans. Am. Math. Soc. 315, 69–87 (1989)Google Scholar
  24. 24.
    Merrill, K.: Smooth well-localized Parseval wavelets based on wavelet sets in \(\mathbb R^2\). Contemp. Math. 464, 161–175 (2008)Google Scholar
  25. 25.
    Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  26. 26.
    Meyer, Y.: Wavelets: Algorithms and Applications. Society for Industrial and Applied Mathematics, Philadelphia (1993)zbMATHGoogle Scholar
  27. 27.
    Paluszyński, M., Šikić, H., Weiss, G., Xiao, S.: Generalized low pass filters and MRA frame wavelets. J. Geom. Anal. 11, 311–342 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Paluszyński, M., Šikić, H., Weiss, G., Xiao, S.: Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties. Adv. Comp. Math. 18, 297–327 (2003)zbMATHGoogle Scholar
  29. 29.
    Papadakis, M.: Generalized frame multiresolution analysis of abstract Hilbert spaces. In: Benedetto, J., Zayed, A. (eds.) Sampling, Wavelets and Tomography, pp. 179–223. Birkhäuser, Boston (2004)CrossRefGoogle Scholar
  30. 30.
    Ron, A., Shen, Z.: Affine systems in \(L^2(\mathbb R^d)\): the analysis of the analysis operator. J. Fourier Anal. Appl. 3, 408–447 (1997)Google Scholar
  31. 31.
    Strichartz, R.S.: How to make wavelets. Am. Math. Mon. 100, 539–556 (1993)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zalik, R.A.: Riesz bases and multiresolution analyses. Appl. Comput. Harmon. Anal. 7, 315–331 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

Personalised recommendations