• They are a superset of the Fourier transform.
  The discrete Fourier transform is at one extreme of a continuum.
• Wavelets are better suited to time-limited data.

• Time-frequency analysis
• Noise removal - smoothing
• Image Compression
• Reducing dimensionality of data for computation speed

• Example: Dirac Delta Function: Comparison of wavelet and fourier transforms
• Another example: Sinewave with jumps; Wavelet analysis
• A crude example of analysis with a simple wavelet:

• The first symmetrical and orthogonal wavelet discovered was a simple square wave, which is not continuous, and is known as the Harr wavelet.
  Harr: Compact, orthogonal, symmetric, discontinuous
  
• Mathematical statement of the conditions for orthogonality by Ingrid Daubechies led to the construction of the continuous wavelets here called Daublets, which are far from symmetric.
  Daublets: Compact, orthogonal, non-symmetric, continuous
  
• These are not unique, and more nearly symmetric continuous wavelets exist.
  Symmlets: Compact, orthogonal, nearly symmetric, continuous
  
• If one demands symmetry, one must give up the requirement of continuity; the harr wavelet is not the only possibility. In addition, one now has to use different functions, called "duals" for the reconstruction.
  Fully Symmetrical Orthogonal Wavelets are Discontinuous

• First pick it apart:
  "Multi-resolution" Decomposition
• Then put it all back together:
  Multi-resolution approximations (composition):

Applications:

• Comparison of two different smoothing methods for a
  Noisy Doppler signal
• The "Waveshrink" method removes the smalles coeficients and shrink those that remain, on the assumption that all coeficients contain the same amount of noise. Here it is applied to
  A Noisy NMR Signal:

• Two dimensional wavelet transform (Ingrid Daubechies):
• Fingerprint storage and reconstruction (top 5% of coefficients)

• Fast algorithms exist (similar to FFT)
• Data compression (reduction of dimensionality)

• Applied Wavelet Analysis with S-Plus
  Andrew Bruce & Hong-Ye Gao Springer 1966
• A Friendly Guide to Wavelets
  Springer
  (Friendly to mathematicians, perhaps)

• Ingrid Daubechies Home Page
• Ingrid Daubechies: Ten Lectures on Wavelets
• The Wavelet Digest an electronic publication edited by Wim Sweldens