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


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.


Orthonormal Basis Trigonometric Polynomial Mother Wavelet Continuous Wavelet Transform Haar Wavelet 
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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

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