• Kenneth R. Davidson
  • Allan P. Donsig
Part of the Undergraduate Texts in Mathematics book series (UTM)


In this chapter we develop an important variation on Fourier series, replacing the sine and cosine functions with new families of functions, called wavelets. The strategy is to construct wavelets so that they have some of the good properties of trig functions but avoid the failings of Fourier series that we have seen in previous chapters. With such functions, we can develop new versions of Fourier series methods that will work well for problems where traditional Fourier series work poorly.

What are the good properties of trig functions? First and foremost, we have an orthogonal basis in L2, namely the set of functions sin(nx) and cos(nx) as n runs over \({\mathbb{N}_0}\). This leads to the idea of breaking up a wave into its harmonic constituents, as the sine and cosine functions appear in the solution of the wave equation. We want to retain some version of this orthogonality.


Fourier Series Orthonormal Basis Compact Support Scaling Function Wavelet Basis 
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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooOntarioCanada
  2. 2.Department of MathematicsUniversity of Nebraska-LincolnLincolnUSA

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