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


The history of wavelets is a story that demonstrates the power of collaboration between different specialties within mathematics, physics, engineering, and computer science. In this chapter, we give a brief outline of this history, focusing on the evolution of the associated multiresolution structures.


  1. 1.
    Akansu, A., Haddad, R.: Multiresolution Signal Decomposition: Transforms, Subbands and Wavelets. Academic Press, San Diego (2001)zbMATHGoogle Scholar
  2. 2.
    Baggett, L., Merrill, K.: Abstract harmonic analysis and wavelets in \(\mathbb R^n\). Contemp. Math. 247, 17–27 (1999)Google Scholar
  3. 3.
    Baggett, L., Carey, A., Moran, W., Ohring, P.: General existence theorems for orthonormal wavelets, an abstract approach. Publ. Res. Inst. Math. Sci. 31, 95–111 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baggett, L., Medina, H., Merrill, K.: Generalized multiresolution analyses and a construction procedure for all wavelet sets in \(\mathbb R^n\). J. Fourier Anal. Appl. 5, 563–573 (1999)Google Scholar
  5. 5.
    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
  6. 6.
    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
  7. 7.
    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
  8. 8.
    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
  9. 9.
    Baggett, L., Furst, V., Merrill, K., Packer, J.: Classification of generalized multiresolution analyses. J. Funct. Anal. 258, 4210–4228 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Bildea, S., Dutkay, D., Picioroaga, G.: MRA super-wavelets. N. Y. J. Math. 11, 1–19 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bohnstengel, J., Kesseböhmer, M.: Wavelets for iterated function systems. J. Funct. Anal. 259, 583–601 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bownik, M., Rzeszotnik, Z.: The spectral function of shift-invariant spaces. Mich. Math. J. 51, 387–414 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bownik, M., Rzeszotnik, Z., Speegle, D.: A characterization of dimension functions of wavelets. Appl. Comput. Harmon. Anal. 10, 71–92 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bratteli, O., Jorgensen, P.: Wavelets Through a Looking Glass. Birkhäuser, Boston (2002)CrossRefGoogle Scholar
  16. 16.
    Dai, X., Larson, D., Speegle, D.: Wavelet sets in \(\mathbb R^n\). J. Fourier Anal. Appl. 3, 451–456 (1997)Google Scholar
  17. 17.
    D’Andrea, J.: Constructing fractal wavelet frames. Numer. Funct. Anal. Optim. 33, 906–927 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    D’Andrea, J., Merrill, K., Packer, J.: Fractal wavelets of Dutkay-Jorgensen type for the Sierpinski gasket space. Contemp. Math. 451, 69–88 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992)CrossRefGoogle Scholar
  21. 21.
    Daubechies, I.: Where do wavelets come from?–A personal point of view. Proc. IEEE 84, 510–513 (1996)CrossRefGoogle Scholar
  22. 22.
    de Boor, C., DeVore, R., Ron, A.: The structure of finitely generated shift-invariant subspaces of \(L^2(\mathbb R^d)\). J. Funct. Anal. 119, 37–78 (1994)Google Scholar
  23. 23.
    Dutkay, D., Jorgensen, P.: Wavelets on fractals. Rev. Math. Iberoamericana 22, 131–180 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Feichtinger, H., Gröchenig, K.: A unified approach to atomic decompositions through integrable group representations. In: Cwikel, M., et al. (eds.) Function Spaces and Applications, pp. 52–73. Springer, Berlin (1988)CrossRefGoogle Scholar
  25. 25.
    Führ, Hartmut: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Springer-Verlag, Berlin, Heidelberg (2005)CrossRefGoogle Scholar
  26. 26.
    Guo, K., Labate, D., Lim, W., Weiss, G., Wilson, E.: The theory of wavelets with composite dilations. In: Heil, C. (ed.) Harmonic Analysis and Applications, pp. 231–250. Birkhäuser, Boston (2006)CrossRefGoogle Scholar
  27. 27.
    Grossmann, A., Morlet, J.: Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. 15, 723–736 (1984)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Helson, H.: Lectures on Invariant Subspaces. Academic Press, New York (1964)zbMATHGoogle Scholar
  30. 30.
    Hernandez, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996)CrossRefGoogle Scholar
  31. 31.
    Larsen, N., Raeburn, I.: From filters to wavelets via direct limits. Contemp. Math. 414, 35–40 (2006)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Larson, D., Massopust, P.: Coxeter groups and wavelet sets. Contemp. Math. 451, 187–218 (2008)MathSciNetCrossRefGoogle Scholar
  33. 33.
    MacArthur, J., Taylor, K.: Wavelets with crystal symmetry shifts. J. Fourier Anal. Appl. 17, 1109–1118 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mallat, S.: Multiresolution approximations and wavelet orthonormal bases of \(L^2(\mathbb R)\). Trans. Am. Math. Soc. 315, 69–87 (1989)Google Scholar
  35. 35.
    Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989)CrossRefGoogle Scholar
  36. 36.
    Massopust, P.: Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, Orlando (1995)zbMATHGoogle Scholar
  37. 37.
    Merrill, K.: Simple wavelet sets for scalar dilations in \(L^2(\mathbb R^2)\). In: Jorgensen, P., Merrill, K., Packer, J. (eds.) Wavelets and Frames: A Celebration of the Mathematical Work of Lawrence Baggett, pp. 177–192. Birkhäuser, Boston (2008)Google Scholar
  38. 38.
    Merrill, K.: Smooth well-localized Parseval wavelets based on wavelet sets in \(\mathbb R^2\). Contemp. Math. 464, 161–175 (2008)Google Scholar
  39. 39.
    Merrill, K.: Simple wavelet sets for matrix dilations in \(\mathbb R^2\). Numer. Funct. Anal. Optim. 33, 1112–1125 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Merrill, K.: Simple wavelet sets in \(\mathbb R^n\). J. Geom. Anal. 25, 1295–1305 (2015)Google Scholar
  41. 41.
    Meyer, Y.: Ondelettes, fonctions splines et analyses graduées. Rapport Ceremade 8703 (1987)Google Scholar
  42. 42.
    Meyer, Y.: Wavelets: Algorithms and Applications. Society for Industrial and Applied Mathematics, Philadelphia (1993)zbMATHGoogle Scholar
  43. 43.
    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
  44. 44.
    Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces of \(L_2(\mathbb R^d)\). Can. J. Math. 47, 1051–1094 (1995)Google Scholar
  45. 45.
    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
  46. 46.
    Ron, A., Shen, Z.: The wavelet dimension function is the trace function of a shift-invariant system. Proc. Am. Math. Soc. 131, 1385–1398 (2002)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Weber, E.: Applications of the wavelet multiplicity function. Contemp. Math. 247, 297–306 (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