The Construction of Wavelet Sets

  • John J. BenedettoEmail author
  • Robert L. Benedetto
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Sets Ω in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1 Ω of the set Ω is a single dyadic orthonormal wavelet. The iterative construction is characterized by its generality, its computational implementation, and its simplicity. The construction is transported to the case of locally compact abelian groups G with compact open subgroups H. The best known example of such a group is \(G = {\mathbb{Q}}_{p}\), the field of p-adic rational numbers (as a group under addition), which has the compact open subgroup \(H = {\mathbb{Z}}_{p}\), the ring of p-adic integers. Fascinating intricacies arise. Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. However, our wavelet theory is formulated on G with new group theoretic operators, which can be thought of as analogues of Euclidean translations. As such, our theory for G is structurally cohesive and of significant generality. For perspective, the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, whereas their analogues for G are equivalent.


Multiresolution Analysis Tight Frame Compact Abelian Group Wavelet Theory Coset Representative 
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.



The first named author gratefully acknowledges support from ONR Grant N0001409103 and MURI-ARO Grant W911NF-09-1-0383. He is also especially appreciative of wonderful mathematical interaction through the years, on the Euclidean aspect of this topic, with Professors Larry Baggett, David Larson, and Kathy Merrill, and for more recent invaluable technical contributions by Dr. Christopher Shaw. The second named author gratefully acknowledges support from NSF DMS Grant 0901494.


  1. 1.
    Patrick R. Ahern and Robert I. Jewett, Factorization of locally compact abelian groups, Illinois J. Math. 9 (1965), 230–235.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Hugo A. Aimar, Ana L. Bernardis, and Osvaldo P. Gorosito, Perturbations of the Haar wavelet by convolution, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3619–3621 (electronic).Google Scholar
  3. 3.
    Syed Twareque Ali, Jean-Pierre Antoine, and Jean-Pierre Gazeau, Coherent States, Wavelets and their Generalizations, Springer-Verlag, New York, 2000.Google Scholar
  4. 4.
    Mikhail V. Altaiski, p-adic wavelet decomposition vs Fourier analysis on spheres, Indian J. Pure Appl. Math. 28 (1997), no. 2, 197–205.Google Scholar
  5. 5.
    Mikhail V. Altaisky, p-adic wavelet transform and quantum physics, Tr. Mat. Inst. Steklova 245 (2004), no. Izbr. Vopr. p-adich. Mat. Fiz. i Anal., 41–46.Google Scholar
  6. 6.
    Jean-Pierre Antoine, Yebeni B. Kouagou, Dominique Lambert, and Bruno Torrésani, An algebraic approach to discrete dilations. Application to discrete wavelet transforms, J. Fourier Anal. Appl. 6 (2000), no. 2, 113–141.Google Scholar
  7. 7.
    Jean-Pierre Antoine and Pierre Vandergheynst, Wavelets on the 2-sphere: a group-theoretical approach, Appl. Comput. Harmon. Anal. 7 (1999), no. 3, 262–291.Google Scholar
  8. 8.
    Pascal Auscher, Solution of two problems on wavelets, J. Geom. Anal. 5 (1995), no. 2, 181–236.Google Scholar
  9. 9.
    Larry Baggett, Alan Carey, William Moran, and Peter Ohring, General existence theorems for orthonormal wavelets, an abstract approach, Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 95–111.Google Scholar
  10. 10.
    Lawrence W. Baggett, Palle E. T. Jorgensen, Kathy D. Merrill, and Judith A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46 (2005), no. 8, 083502, 28.Google Scholar
  11. 11.
    Lawrence W. Baggett, Herbert A. Medina, and Kathy D. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℝ n, J. Fourier Anal. Appl. 5 (1999), no. 6, 563–573.Google Scholar
  12. 12.
    Lawrence W. Baggett and Kathy D. Merrill, Abstract harmonic analysis and wavelets in ℝ n, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), Amer. Math. Soc., Providence, RI, 1999, pp. 17–27.Google Scholar
  13. 13.
    John J. Benedetto, Idele characters in spectral synthesis on ℝ∕2πℤ, Ann. Inst. Fourier 23 (1973), 43–64.MathSciNetGoogle Scholar
  14. 14.
    ——, Zeta functions for idelic pseudo-measures, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1979), no. 6, 367–377.Google Scholar
  15. 15.
    ——, Harmonic Analysis and Applications, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1997.Google Scholar
  16. 16.
    ——, Frames, sampling, and seizure prediction, Advances in Wavelets, edited by K.-S. Lau, Springer-Verlag, New York, 1998.Google Scholar
  17. 17.
    John J. Benedetto and Robert L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal. 14 (2004), 423–456.MathSciNetzbMATHGoogle Scholar
  18. 18.
    John J. Benedetto and Wojciech Czaja, Integration and Modern Analysis, Birkhäuser Advanced texts, Birkhäuser Boston, MA, 2009.zbMATHCrossRefGoogle Scholar
  19. 19.
    John J. Benedetto and Emily J. King, Smooth functions associated with wavelet sets on ℝ d , d ≥ 1, and frame bound gaps, Acta Appl. Math. 107 (2009), no. 1-3, 121–142.Google Scholar
  20. 20.
    John J. Benedetto and Manuel T. Leon, The construction of multiple dyadic minimally supported frequency wavelets on ℝ d, Contemporary Mathematics 247 (1999), 43–74.MathSciNetGoogle Scholar
  21. 21.
    ——, The construction of single wavelets in d-dimensions, J. of Geometric Analysis 11 (2001), 1–15.Google Scholar
  22. 22.
    John J. Benedetto and Songkiat Sumetkijakan, A fractal set constructed from a class of wavelet sets, Inverse problems, image analysis, and medical imaging (New Orleans, LA, 2001), Contemp. Math., vol. 313, Amer. Math. Soc., Providence, RI, 2002, pp. 19–35.Google Scholar
  23. 23.
    ——, Tight frames and geometric properties of wavelet sets, Adv. Comput. Math. 24 (2006), no. 1-4, 35–56.Google Scholar
  24. 24.
    Robert L. Benedetto, Examples of wavelets for local fields, Wavelets, Frames, and Operator Theory (College Park, MD, 2003), Amer. Math. Soc., Providence, RI, 2004, pp. 27–47.Google Scholar
  25. 25.
    Andrea Calogero, A characterization of wavelets on general lattices, J. Geom. Anal. 10 (2000), no. 4, 597–622.Google Scholar
  26. 26.
    Ole Christensen, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 273–291 (electronic).Google Scholar
  27. 27.
    Charles K. Chui, Wojciech Czaja, Mauro Maggioni, and Guido Weiss, Characterization of general tight frames with matrix dilations and tightness preserving oversampling, J. Fourier Anal. Appl. 8 (2002), no. 2, 173–200.Google Scholar
  28. 28.
    Albert Cohen and Ingrid Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), no. 1, 51–137.Google Scholar
  29. 29.
    Ronald R. Coifman and Guido Weiss, Book Review: Littlewood-Paley and Multiplier Theory, Bull. Amer. Math. Soc. 84 (1978), no. 2, 242–250.Google Scholar
  30. 30.
    Jennifer Courter, Construction of dilation-d wavelets, AMS Contemporary Mathematics 247 (1999), 183–205.MathSciNetGoogle Scholar
  31. 31.
    Stephan Dahlke, Multiresolution analysis and wavelets on locally compact abelian groups, Wavelets, images, and surface fitting (Chamonix-Mont-Blanc, 1993), A K Peters, Wellesley, MA, 1994, pp. 141–156.Google Scholar
  32. 32.
    ——, The construction of wavelets on groups and manifolds, General algebra and discrete mathematics (Potsdam, 1993), Heldermann, Lemgo, 1995, pp. 47–58.Google Scholar
  33. 33.
    Xingde Dai, Yuanan Diao, Qing Gu, and Deguang Han, Frame wavelets in subspaces of L 2 (ℝ d), Proc. Amer. Math. Soc. 130 (2002), no. 11, 3259–3267 (electronic).Google Scholar
  34. 34.
    Xingde Dai and David R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), no. 640, viii+68.Google Scholar
  35. 35.
    Xingde Dai, David R. Larson, and Darrin M. Speegle, Wavelet sets in ℝ n, J. Fourier Anal. Appl. 3 (1997), no. 4, 451–456.Google Scholar
  36. 36.
    ——, Wavelet sets in ℝ n . II, Wavelets, multiwavelets, and their applications (San Diego, CA, 1997), Contemp. Math., vol. 216, Amer. Math. Soc., Providence, RI, 1998, pp. 15–40.Google Scholar
  37. 37.
    James Daly and Keith Phillips, On singular integrals, multipliers, ℌ 1 and Fourier series — a local field phenomenon, Math. Ann. 265 (1983), no. 2, 181–219.Google Scholar
  38. 38.
    Ingrid Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Math. SIAM, Philadelphia, PA, 1992.zbMATHGoogle Scholar
  39. 39.
    Ingrid Daubechies and Bin Han, The canonical dual frame of a wavelet frame, Appl. Comput. Harmon. Anal. 12 (2002), no. 3, 269–285.Google Scholar
  40. 40.
    Richard J. Duffin and Albert C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.MathSciNetzbMATHGoogle Scholar
  41. 41.
    Robert E. Edwards and Garth I. Gaudry, Littlewood-Paley and multiplier theory, Springer-Verlag, Berlin, 1977, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90.Google Scholar
  42. 42.
    Xiang Fang and Xihua Wang, Construction of minimally supported frequency wavelets, J. Fourier Anal. Appl. 2 (1996), no. 4, 315–327.Google Scholar
  43. 43.
    Yuri A. Farkov, Orthogonal wavelets on locally compact abelian groups, Funktsional. Anal. i Prilozhen. 31 (1997), no. 4, 86–88.Google Scholar
  44. 44.
    ——, Orthogonal p-wavelets on ℝ +, Wavelets and Splines, St. Petersburg Univ. Press, St. Petersburg, 2005, pp. 4–26.Google Scholar
  45. 45.
    ——, Orthogonal wavelets with compact supports on locally compact abelian groups, Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005), no. 3, 193–220.Google Scholar
  46. 46.
    Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Amer. Math. Soc., Washington, DC, 1991.Google Scholar
  47. 47.
    Gustaf Gripenberg, A necessary and sufficient condition for the existence of a father wavelet, Studia Math. 114 (1995), no. 3, 207–226.Google Scholar
  48. 48.
    Karlheinz Gröchenig and Wolodymyr R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of L 2 (ℝ n), IEEE Trans. Information Theory 38(2) (1992), 556–568.Google Scholar
  49. 49.
    Qing Gu and Deguang Han, On multiresolution analysis (MRA) wavelets in ℝ n, J. Fourier Anal. Appl. 6 (2000), no. 4, 437–447.Google Scholar
  50. 50.
    Bin Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997), no. 4, 380–413.Google Scholar
  51. 51.
    Deguang Han, Unitary systems, wavelets, and operator algebras, Ph.D. thesis, Texas A&M University, College Station, TX, 1997.Google Scholar
  52. 52.
    Deguang Han, David R. Larson, Manos Papadakis, and Theodoros Stavropoulos, Multiresolution analyses of abstract Hilbert spaces and wandering subspaces, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), Amer. Math. Soc., Providence, RI, 1999, pp. 259–284.Google Scholar
  53. 53.
    Eugenio Hernández, Xihua Wang, and Guido Weiss, Smoothing minimally supported frequency wavelets. I, J. Fourier Anal. Appl. 2 (1996), no. 4, 329–340.Google Scholar
  54. 54.
    ——, Smoothing minimally supported frequency wavelets, part II, J. Fourier Anal. Appl. 3 (1997), 23–41.Google Scholar
  55. 55.
    Eugenio Hernández and Guido Weiss, A First Course on Wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996, With a foreword by Yves Meyer.Google Scholar
  56. 56.
    Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Volume I, Springer-Verlag, New York, 1963.Google Scholar
  57. 57.
    ——, Abstract Harmonic Analysis, Volume II, Springer-Verlag, New York, 1970.Google Scholar
  58. 58.
    Matthias Holschneider, Wavelet analysis over abelian groups, Appl. Comput. Harmon. Anal. 2 (1995), no. 1, 52–60.Google Scholar
  59. 59.
    Huikun Jiang, Dengfeng Li, and Ning Jin, Multiresolution analysis on local fields, J. Math. Anal. Appl. 294 (2004), no. 2, 523–532.Google Scholar
  60. 60.
    Carolyn P. Johnston, On the pseudodilation representations of Flornes, Grossmann, Holschneider, and Torresani, J. Fourier Anal. Appl. 3 (1997), no. 4, 377–385.Google Scholar
  61. 61.
    Andrei Yu. Khrennikov and Sergei V. Kozyrev, Wavelets on ultrametric spaces, Appl. Comput. Harmon. Anal. 19 (2005), no. 1, 61–76.Google Scholar
  62. 62.
    Andrei Yu. Khrennikov, Vladimir M. Shelkovich, and Maria Skopina, p-adic refinable functions and MRA-based wavelets, J. Approx. Theory 161 (2009), no. 1, 226–238.Google Scholar
  63. 63.
    Andrei Kolmogorov, Une contribution à l’étude de la convergence des séries de Fourier, Fund. Math. 5 (1924), 96–97.Google Scholar
  64. 64.
    Jelena Kovačević and Martin Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for ℛ n, IEEE Trans. Inform. Theory 38 (1992), no. 2, part 2, 533–555.Google Scholar
  65. 65.
    Sergei V. Kozyrev, Wavelet analysis as a p-adic spectral analysis, Izv. Math 66 (2002), 367–376.MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    ——, p-adic pseudodifferential operators and p-adic wavelets, Theoret. and Math. Phys. 138 (2004), no. 3, 322–332.Google Scholar
  67. 67.
    Jeffrey C. Lagarias and Yang Wang, Integral self-affine tiles in ℝ n , part II lattice tilings, J. Fourier Anal. Appl. 3 (1997), 83–102.MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Jeffrey C. Lagarias and Yang Wang, Orthogonality criteria for compactly supported refinable functions and refinable function vectors, J. Fourier Anal. Appl. 6 (2000), no. 2, 153–170.Google Scholar
  69. 69.
    W. Christopher Lang, Orthogonal wavelets on the Cantor dyadic group, SIAM J. Math. Anal. 27 (1996), 305–312.MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    ——, Fractal multiwavelets related to the Cantor dyadic group, Internat. J. Math. Math. Sci. 21 (1998), 307–314.Google Scholar
  71. 71.
    ——, Wavelet analysis on the Cantor dyadic group, Houston J. Math. 24 (1998), 533–544 and 757–758.Google Scholar
  72. 72.
    Pierre G. Lemarié, Base d’ondelettes sur les groupes de Lie stratifiés, Bull. Soc. Math. France 117 (1989), no. 2, 211–232.Google Scholar
  73. 73.
    Yunzhang Li, On the holes of a class of bidimensional nonseparable wavelets, J. Approx. Theory 125 (2003), no. 2, 151–168.Google Scholar
  74. 74.
    John E. Littlewood and Raymond E. A. C. Paley, Theorems on Fourier series and power series, J. London Math. Soc. 6 (1931), 230–233.Google Scholar
  75. 75.
    ——, Theorems on Fourier series and power series III, Proc. London Math. Soc. 43 (1937), 105–126.Google Scholar
  76. 76.
    Taras P. Lukashenko, Wavelets on topological groups, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 3, 88–102.Google Scholar
  77. 77.
    Wolodymyr R. Madych, Some elementary properties of multiresolution analyses of L 2 (ℝ n), Wavelets: a Tutorial in Theory and Applications, edited by Charles K. Chui, Academic Press, Inc, San Diego, CA, 1992, pp. 256–294.Google Scholar
  78. 78.
    Stéphane G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Boston, 1998.zbMATHGoogle Scholar
  79. 79.
    Kathy D. Merrill, Simple wavelet sets for integral dilations in ℝ 2, Representations, Wavelets, and Frames: A Celebration of the Mathematical Work of Lawrence W. Baggett, Birkhäuser, Boston, MA, 2008, pp. 177–192.Google Scholar
  80. 80.
    Yves Meyer, Ondelettes, fonctions splines et analyses graduées, Rend. Sem. Mat. Univ. Politec. Torino 45 (1987), no. 1, 1–42 (1988).Google Scholar
  81. 81.
    ——, Ondelettes et Opérateurs, Hermann, Paris, 1990.Google Scholar
  82. 82.
    ——, Wavelets and Operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992, Translated from the 1990 French original by D. H. Salinger.Google Scholar
  83. 83.
    Cui Minggen, GuangHong Gao, and Phil Ung Chung, On the wavelet transform in the field ℚ p of p-adic numbers, Appl. Comput. Harmon. Anal. 13 (2002), no. 2, 162–168.Google Scholar
  84. 84.
    Manos Papadakis, Generalized frame multiresolution analysis of abstract Hilbert spaces, Sampling, Wavelets, and Tomography, Birkhäuser, Boston, MA, 2003.Google Scholar
  85. 85.
    Manos Papadakis and Theodoros Stavropoulos, On the multiresolution analyses of abstract Hilbert spaces, Bull. Greek Math. Soc. 40 (1998), 79–92.MathSciNetGoogle Scholar
  86. 86.
    Lev Lemenovich Pontryagin, Topological Groups, 2nd edition, Gordon and Breach, Science Publishers, Inc., New York, 1966, Translated from the Russian by Arlen Brown.Google Scholar
  87. 87.
    Hans Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, 1968.Google Scholar
  88. 88.
    Walter Rudin, Fourier Analysis on Groups, John Wiley and Sons, New York, 1962.zbMATHGoogle Scholar
  89. 89.
    Eckart Schulz and Keith F. Taylor, Extensions of the Heisenberg group and wavelet analysis in the plane, Spline functions and the theory of wavelets (Montreal, PQ, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 217–225. MR99m:42053Google Scholar
  90. 90.
    Vladimir Shelkovich and Maria Skopina, p-adic Haar multiresolution analysis and pseudo-differential operators, J. Fourier Anal. Appl. 15 (2009), no. 3, 366–393.Google Scholar
  91. 91.
    Paolo M. Soardi and David Weiland, Single wavelets in n-dimensions, J. Fourier Anal. Appl. 4 (1998), 299–315.MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.Google Scholar
  93. 93.
    ——, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J., 1970.Google Scholar
  94. 94.
    Elias M. Stein and Guido Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971, Princeton Mathematical Series, No. 32.Google Scholar
  95. 95.
    Robert S. Strichartz, Construction of orthonormal wavelets, Wavelets: Mathematics and Applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 23–50.Google Scholar
  96. 96.
    Mitchell H. Taibleson, Fourier Analysis on Local Fields, Princeton U. Press, Princeton, NJ, 1975.zbMATHGoogle Scholar
  97. 97.
    Khalifa Trimèche, Continuous wavelet transform on semisimple Lie groups and inversion of the Abel transform and its dual, Collect. Math. 47 (1996), no. 3, 231–268.Google Scholar
  98. 98.
    Victor G. Zakharov, Nonseparable multidimensional Littlewood-Paley like wavelet bases, Preprint, 1996.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations