Advertisement

Wavelets

  • George Bachman
  • Lawrence Narici
  • Edward Beckenstein
Chapter
  • 1.1k Downloads
Part of the Universitext book series (UTX)

Abstract

By a Dilation of f(t) we mean f (kt) for some constant k.When we considered Fourier analysis in L 2[-π, π], we made extensive use of the fact that normalized dilations (l/√π) e int, nZ, of e itconstitute an orthonormal basis for L 2[π, π]. The procedure permits any function fL 2 [— π, π] to be represented as an infinite series of complex exponentials in the sense of convergence to f in the L 2-norm. As there are other orthonormal bases for L 2 [—π, π], it would be profligate to ignore them. Wavelet analysis not only employs orthonormal bases other than (l/√2π) e int ,it uses bases that are not orthonormal (Riesz bases) and basis-like collections (frames) that are not even linearly independent. In some cases (fingerprints) wavelet analysis is much better than Fourier analysis in the sense that fewer terms suffice to approximate certain functions.

Keywords

Orthonormal Basis Trigonometric Polynomial Mother Wavelet Continuous Wavelet Transform Haar Wavelet 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Aldroubi 1995, Portraits of frames, Proc. Amer. Math. Soc. 123, 1661–1668.MathSciNetzbMATHGoogle Scholar
  2. 2.
    J.M. Ash 1989, Uniqueness of representation by trigonometric series, Amer. Math. Monthly 96, 873–885.MathSciNetzbMATHGoogle Scholar
  3. 3.
    R. Ash and M. Gardner 1975, Topics in stochastic processes, Academic Press, New York.zbMATHGoogle Scholar
  4. 4.
    L. Auslander, T. Kailath, and S. Mitter (editors) 1990, Signal processing theory, Springer-Verlag, New York; see especially Wavelets and frames by C. Heil}.Google Scholar
  5. 5.
    G. Bachman and L. Narici 1964, Elements of abstract harmonic analysis, Academic Press, New York.zbMATHGoogle Scholar
  6. 6.
    G. Bachman and L. Narici 1966. Functional analysis, Academic Press, New York. Reprinted by Dover Publications, Mineola, NY, 199Google Scholar
  7. 7.
    S. Barnett 1990, Matrices—methods and applications, Clarendon Press, Oxford.Google Scholar
  8. 8.
    R. Bartle 1995, The elements of integration and Lebesgue measure, Wiley Classics Library, New York.zbMATHCrossRefGoogle Scholar
  9. 9.
    N.K. Bary 1964, A treatise on trigonometric series I, Macmillan, New York.zbMATHGoogle Scholar
  10. 10.
    J. Benedetto 1997, Harmonic analysis and applications, CRC Press, Boca Raton.Google Scholar
  11. 11.
    J. Benedetto and — Frazier 1993, Wavelets: Mathematics and applications, CRC Press, Boca RatonGoogle Scholar
  12. 12.
    G. Bergland 1967, The fast Fourier transform recursive equations for arbitrary length records, Math. Comp. 21, 236–238.MathSciNetzbMATHGoogle Scholar
  13. 13.
    F. Beutler 1961, Sampling theorems and bases in — Hilbert space, Information and Control 4, 97–117.MathSciNetzbMATHGoogle Scholar
  14. 14.
    M. Bôcher, 1906, Introduction to the theory of Fourier’ — series, Annals of Math. 7.Google Scholar
  15. 15.
    S. Bochner and K. Chandrasekharan 1949, Fourier transforms, Annals of Mathematics Studies 19, Princeton University Press, Princeton, NJ.zbMATHGoogle Scholar
  16. 16.
    O. Bonnet, 1850, Mémoires couronnés et mémoires des savants Etrangers publiés par l’ académie royale des sciences, des lettres et des beaux-arts de Belgique, 23.Google Scholar
  17. 17.
    D. Bressoud 1994, A radical approach to real analysis, The Mathematical Association of America, Washington, D.C. Uses Fourier series as basis for exploring analysis with good historical notes.Google Scholar
  18. 18.
    W. Briggs and V.E. Henson 1995, The DFT, an owner’ — manual for the discrete Fourier transform, SIAM.Google Scholar
  19. 19.
    G. Cantor 1870, Beweis, das eine für jeden reellen Wert von — durch eine trigonometrische Reihe gegebene Funktion — (x) sich nur auf eine einzige Weise in dieser Form darstellen lässt, J. f, d. reine u. angew. Math. 72, 139–142; also in Gesammelte Abhandlungen, Georg Olms, Hildesheim 1962, 80-83.zbMATHGoogle Scholar
  20. 20.
    L. Carleson 1966, On convergence and growth of partial sums of Fourier series, Acta Math. 116, 135–157.MathSciNetzbMATHGoogle Scholar
  21. 21.
    L. Carslaw 1930, Introduction to the theory of Fourier’ — series and integrals, Macmillan, New York; reprinted by Dover, New York 1952. Thorough and readable.Google Scholar
  22. 22.
    M. Cartwright 1990, Fourier methods, Ellis Horwood, Chichester.zbMATHGoogle Scholar
  23. 23.
    A.L. Cauchy 1841, Mémoire sur diverses formulaes de analyse, C. R. Acad. Sci. Paris 12, 283–298.Google Scholar
  24. 24.
    C. Chui 1992a, An introduction to wavelets, Academic Press, San Diego.zbMATHGoogle Scholar
  25. 25.
    C. Chui 1992b, Wavelets: — tutorial in theory and applications, Academic Press, San Diego.zbMATHGoogle Scholar
  26. 26.
    B. Cipra 1993, Wavelet applications come to the fore, SIAM News, November.Google Scholar
  27. 27.
    R. Cooke 1979, The Cantor-Lebesgue theorem, Amer. Math. Monthly 86, 558–565.MathSciNetzbMATHGoogle Scholar
  28. 28.
    J. Cooley and J. Tukey 1965, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19, 297–301.MathSciNetzbMATHGoogle Scholar
  29. 29.
    J. Corrado 1984, Vibration signatures are easy to read with FFT, Design News 1/23/84, 125–128. How the FFT is used to eliminate vibrations in machinery.Google Scholar
  30. 30.
    R. Courant and H. Robbins 1941, What is mathematics? Oxford University Press, New York.Google Scholar
  31. 31.
    I. Daubechies 1988, Orihonormal bases of compactly supported wavelets, Comm, Pure and Applied Math. 41, 909–996.MathSciNetzbMATHGoogle Scholar
  32. 32.
    I. Daubechies 1991, The wavelet transform: — method of time-frequency localization in Advances in spectrum analysis and array processing 1, edited by S. Haykin, Prentice-Hall, Englewood Cliffs.Google Scholar
  33. 33.
    I. Daubechies 1992, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.zbMATHCrossRefGoogle Scholar
  34. 34.
    I. Daubechies, A. Grossman and Y. Meyer 1986, Painless nonorthogonal expansions, J. Math. Phys. 27, 1271–1283.zbMATHGoogle Scholar
  35. 35.
    P. Davis 1979, Circulant matrices, Wiley, New York.zbMATHGoogle Scholar
  36. 36.
    Ch. De La Vallée-Poussin 1912, Sur Vunicite du developpement trigonometrique, Bull. Acad. Roy. de Belg. 702–718.Google Scholar
  37. 37.
    J. Diestel 1984, Sequences and series in Banach spaces, Springer-Verlag, New York.CrossRefGoogle Scholar
  38. 38.
    P. Dirichlet 1829, Sur la convergence des séries trigonometriques qui servent — représenter une function arbitraire entre des limites donnees, J. für d. reine u. angewandte Mathematik. 4, 157–169.zbMATHGoogle Scholar
  39. 39.
    A. Dvoretzky and C. Rogers 1950, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A., 36, 192–197.Google Scholar
  40. 40.
    N. Dunford and J. Schwartz 1958. Linear operators, Part I, Interscience, New York.Google Scholar
  41. 41.
    H. Dym and H. McKean 1972, Fourier series and integrals, Academic Press, San Diego.zbMATHGoogle Scholar
  42. 42.
    R. Edwards 1967, Fourier series Holt, Rinehart and Winston, New York.Google Scholar
  43. 43.
    H. Feichtinger and — Gröchen 1992, Gabor wavelets and the Heisenberg group in Wavelets: — tutorial in theory and applications edited by C. Chui, Academic Press, San Diego.Google Scholar
  44. 44.
    L. Fejér 1904, Untersuchungen über Fouriersche Reihen, Mathematische Annalen. 58, 51–69.zbMATHGoogle Scholar
  45. 45.
    G. Folland 1995, A course in abstract harmonic analysis, CRC Press, Boca Raton.zbMATHGoogle Scholar
  46. 46.
    J. Fourier, The analytical theory of heat, Dover, Mineola, New York, — translation of the 1822 version of Théorie analytique de la chaleur published by Didot, Paris. The 1822 version is an improvement of Fourier’ — 1811 version. Theorie de la propagation de la chaleur dans les solides, reproduced in Grattan-Guiness 1972.Google Scholar
  47. 47.
    W. Gentlemean and G. Sande 1966, Fast Fourier transform for fun and profit, Proc. 1966 Fall Joint Computer Conference AFIPS 29, 563–578.Google Scholar
  48. 48.
    A. Gluchoff 1994, Trigonometric series and theories of integration, Mathematics Magazine 67, 3–20. How the integrals of Cauchy (continuous functions), Riemann (continuous a.e.), and Lebesgue were invented to address problems concerning Fourier series and their applications to those problems immediately afterward.MathSciNetzbMATHGoogle Scholar
  49. 49.
    R. Goldberg 1976, Methods of real analysis, Wiley, New York.zbMATHGoogle Scholar
  50. 50.
    I. Grattan-Guiness 1972, Joseph Fourier, 1768-1830; — survey of his life and work, MIT Press, Cambridge, MA. Fourier’ — original 1807 paper with excellent annotation.Google Scholar
  51. 51.
    A. Grossman and J. Morlet 1984, Decomposition of Hardy functions into square-integrable wavelets of constant shape, SIAM J. Math. Anal. 15, 723–736.Google Scholar
  52. 52.
    A. Grothendieck 1955, Produits tensoriels topologiques et espaces nucleaires, Memoirs of the American Mathematical Society 16, Providence, R.I.Google Scholar
  53. 53.
    M. Heideman, D. Johnson, and C. Burrus 1984, Gauss and the history of the fast Fourier transform IEE ASSP Mag. (October) 14–21; also in Arch. Hist. Exact Sciences 34 (1985) 265-277.Google Scholar
  54. 54.
    C. Heil and D. Walnut 1989, Continuous and discrete wavelet transforms, SIAM Review 31, 628–666.MathSciNetzbMATHGoogle Scholar
  55. 55.
    E. Hernández and G. Weiss 1996, A first course on wavelets, CRC Press, Boca Raton.zbMATHCrossRefGoogle Scholar
  56. 56.
    J.R. Higgins 1996, Sampling theory in Fourier and signal analysis, Oxford University Press, New York.zbMATHGoogle Scholar
  57. 57.
    S. Hildebrandt and A. Tromba 1985, Mathematics and optimal form, Freeman, New York.Google Scholar
  58. 58.
    E. Hobson 1957, The theory of functions of — real variable, Dover, New York.Google Scholar
  59. 59.
    M. Holschneider 1995, Wavelets: An analysis tool, Oxford, London.Google Scholar
  60. 60.
    R. Hunt 1968, Orthogonal expansions and their continuous analogues, Southern Illinois University Press, Carbondale, Illinois, 235–255.Google Scholar
  61. 61.
    E. Jahnke, F. Emde, and F. Lösch 1960, Tables of higher functions, 6th ed., McGraw-Hill, New York.zbMATHGoogle Scholar
  62. 62.
    O. Jørsboe and L. Mejlbro 1982, The Carleson-Hunt theorem on Fourier series, Lecture Notes in Mathematics 911, Springer-Verlag, New York.zbMATHGoogle Scholar
  63. 63.
    G. Kaiser 1994, A friendly guide to wavelets, Birkhauser, Boston.Google Scholar
  64. 64.
    Y. Katznelson 1968, An introduction to harmonic analysis, Wiley, New York.zbMATHGoogle Scholar
  65. 65.
    T. Kawata 1972, Fourier analysis in probability theory, Academic Press, New York.zbMATHGoogle Scholar
  66. 66.
    H. Koornwlnder 1993, Wavelets: An elementary treatment of theory and applications, edited by T. Koornwinder, World Scientific Press.Google Scholar
  67. 67.
    T. Körner 1988 (reprinted 1992), Fourier analysis, Cambridge University Press, Cambridge-New York.zbMATHGoogle Scholar
  68. 68.
    J. Lindenstrauss and L. Tzafriri 1977, Classical Banach spaces I, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  69. F. Lukács 1920, Über die Berstimmung des Sprunges einer Funktio Google Scholar
  70. aus ihrer Fourierreihe}}, J. für Mathematik 150, 107–112.Google Scholar
  71. 70.
    S. Mallat 1989, Multiresolution approximation and wavelet orthonormal bases of L 2 (R), Trans. Amer. Math. Soc. 315, 69–88.MathSciNetzbMATHGoogle Scholar
  72. 71.
    J. Marsden and M. Hoffman 1993. Elementary classical analysis, W. H. Freeman, New York.zbMATHGoogle Scholar
  73. 72.
    E. McShane 1947, Integration, Princeton University Press, Princeton, NJ.zbMATHGoogle Scholar
  74. 73.
    Y. Meyer 1990, Ondelettes et operations — and II, Hermann, Paris.Google Scholar
  75. 74.
    Y. Meyer 1993, Wavelets—algorithms and applications, (translated and revised by R. Ryan) SIAM.Google Scholar
  76. 75.
    Y. Meyer 1993, Book review, Bull. Amer. Math. Soc. 28, 350–360.Google Scholar
  77. 76.
    N. Morrison 1994, Introduction to Fourier analysis, Wiley, New York.zbMATHGoogle Scholar
  78. 77.
    C. Mozzochi 1971, On the pointwise convergence of Fourier series, Lecture Notes in Mathematics 199, Springer-Verlag, New York.zbMATHGoogle Scholar
  79. 78.
    L. Narici and E. Beckenstein 1985. Topological vector spaces, Marcel Dekker, New YorkGoogle Scholar
  80. 79.
    I. Natanson 1961, Theory of functions of — real variable — and II, Ungar, New York.Google Scholar
  81. 80.
    A. Naylor and G. Sell 1982. Linear operator theory in engineering and science, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  82. 81.
    S. Nikolsky 1977, A course of mathematical analysis, Mir Publishers, Moscow.zbMATHGoogle Scholar
  83. 82.
    M. Plancherel 1924-1925 Le développement de la theorie des series trigonométriques dans le dernier quart de siècle, L’ Enseignement Math. 24.Google Scholar
  84. 83.
    L. Prasad and S. Iyengyar 1997, Wavelet analysis with applications to image processing, CRC Press.Google Scholar
  85. 84.
    C. Rees, S. Shah, and C. Stanojevic 1981, Theory and applications of Fourier series, Marcel Dekker, New York.Google Scholar
  86. 85.
    C. Reind and T. Passin 1992, Signal processing in C, Wiley, New York.Google Scholar
  87. 86.
    B. Riemann 1867, Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe, Göttingen, Abh. Ges. Wiss. 13.Google Scholar
  88. 87.
    H. Royden 1968, Real analysis, 2nd ed., Macmillan, New York.Google Scholar
  89. 88.
    W. Rudin 1962, Fourier analysis on groups, Wiley-Interscience, New York.zbMATHGoogle Scholar
  90. 89.
    W. Rudin 1974. Real and complex analysis, 2nd ed., McGraw-Hill, New York.zbMATHGoogle Scholar
  91. 90.
    W. Rudin 1976, Principles of mathematical analysis, 3rd ed., McGraw-Hill, New York.zbMATHGoogle Scholar
  92. 91.
    M. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and L. Raphael, eds. 1992, Wavelets and their applications, Jones and Bartlett, Boston.Google Scholar
  93. 92.
    G. Sansone 1958, Orthogonal functions, Wiley, New York.Google Scholar
  94. 93.
    G. Sansone and I. Gerretson 1960, Lectures on the theory of functions of — complex variable, Noordhoff, Groningen.Google Scholar
  95. 94.
    C. Shannon 1948, A mathematical theory of communication, Bell System Technical Journal.Google Scholar
  96. 95.
    G. Simmons 1963. Introduction to topology and modern analysis, McGraw-Hill, New York.zbMATHGoogle Scholar
  97. 96.
    G. Strang 1993, Wavelet transforms versus Fourier transforms, Bull. Amer. Math. Soc. 28, 288–305.MathSciNetzbMATHGoogle Scholar
  98. 97.
    R. Strichartz 1993, How to make wavelets, Amer. Math. Monthly 100, 539–556.MathSciNetzbMATHGoogle Scholar
  99. 98.
    K. Stromberg 1981. An introduction to classical real analysis, Wadsworth, Belmont, California.zbMATHGoogle Scholar
  100. 99.
    Z. Szmydt 1977, Fourier transformation and linear differential equations, Reidel Publishing Co., Boston.zbMATHGoogle Scholar
  101. 100.
    B. Sz.-Nagy 1963 Introduction to real functions and orthogonal expansions, Oxford University Press, Oxford.Google Scholar
  102. 101.
    C. Swartz 1992, An introduction to functional analysis, Marcel Dekker, New York.zbMATHGoogle Scholar
  103. 102.
    A. Taylor and W. Mann 1972, Advanced calculus 2nd ed., Xerox Publishing Co., Lexington, MA.zbMATHGoogle Scholar
  104. 103.
    E.C. Titchmarsh 1939, The theory of functions, 2nd ed., Oxford University Press, Oxford.zbMATHGoogle Scholar
  105. 104.
    G.P. Tolstov 1976, Fourier series (translated by R. Silverman), Dover, New York. This is — reprint of the 1962 version published by Prentice-Hall, Englewood Cliffs, New Jersey.zbMATHGoogle Scholar
  106. 105.
    C. Van Loan 1992, Computational frameworks for the fast Fourier transform, SIAM.Google Scholar
  107. 106.
    E.B. Van Vleck 1914, The influence of Fourier’s series upon the development of mathematics, Science 39, no. 995, 113–124. An interesting informative article. In an illustration of how times have changed, another article in the same issue of Science is “Scripture on Stuttering and Speech.”Google Scholar
  108. 107.
    J. Walker 1991, Fast Fourier transforms, CRC Press, Boca Raton.zbMATHGoogle Scholar
  109. 108.
    J. Walker 1997, Fourier analysis and wavelet analysis, Notices Amer. Math. Soc. 44, 658–670.zbMATHGoogle Scholar
  110. 109.
    G. Walter 1994, Wavelet and other orthogonal systems with applications, CRC Press, Boca Raton.Google Scholar
  111. 110.
    R.L. Wheeden and A. Zygmund 1977, Measure and integral, Dekker, New York.zbMATHGoogle Scholar
  112. 111.
    E.T. Whittaker 1915, On the functions which are represented by the expansions of the interpolation theory, Proc. Roy. Soc. Edinburgh, Sec. — (35), 181–194.Google Scholar
  113. 112.
    P. Wojtaszczyk 1997, A mathematical introduction to wavelets, Cambridge University Press, London.zbMATHCrossRefGoogle Scholar
  114. 113.
    R. Young 1980, An introduction to nonharmonic Fourier series, Academic Press, San DiegoGoogle Scholar
  115. 114.
    A. Zygmund 1959, Trigonometric series, 2nd ed., Cambridge University Press, New York.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • George Bachman
    • 1
  • Lawrence Narici
    • 2
  • Edward Beckenstein
    • 3
  1. 1.Emeritus of MathematicsPolytechnic UniversityBrooklynUSA
  2. 2.Department of Mathematics and Computer ScienceSt. John’s UniversityJamaicaUSA
  3. 3.Science DivisionSt. John’s UniversityStaten IslandUSA

Personalised recommendations