The fast wavelet transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain. This algorithm was introduced in 1989 by Stéphane Mallat.
 It has as theoretical foundation the device of a finitely generated, orthogonal multiresolution analysis (MRA). In the terms given there, one selects a sampling scale J with sampling rate of 2J per unit interval, and projects the given signal f onto the space 
  
    
      
        
          V
          
            J
          
        
      
    
    {\displaystyle V_{J}}
  
; in theory by computing the scalar products 
 where 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 is the scaling function of the chosen wavelet transform; in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so
 is the orthogonal projection or at least some good approximation of the original signal in 
  
    
      
        
          V
          
            J
          
        
      
    
    {\displaystyle V_{J}}
  
. 
 The MRA is characterised by its scaling sequence
 and its wavelet sequence 
 (some coefficients might be zero). Those allow to compute the wavelet coefficients 
  
    
      
        
          d
          
            n
          
          
            (
            k
            )
          
        
      
    
    {\displaystyle d_{n}^{(k)}}
  
, at least some range k=M,...,J-1, without having to approximate the integrals in the corresponding scalar products. Instead, one can directly, with the help of convolution and decimation operators, compute those coefficients from the first approximation 
  
    
      
        
          s
          
            (
            J
            )
          
        
      
    
    {\displaystyle s^{(J)}}
  
.
 For the discrete wavelet transform (DWT), 
one computes recursively, starting with the coefficient sequence 
  
    
      
        
          s
          
            (
            J
            )
          
        
      
    
    {\displaystyle s^{(J)}}
  
 and counting down from k = J - 1 to some M < J,
 and 
 for k=J-1,J-2,...,M and all 
  
    
      
        n
        ∈
        
          Z
        
      
    
    {\displaystyle n\in \mathbb {Z} }
  
. In the Z-transform notation:
 It follows that
 is the orthogonal projection of the original signal f or at least of the first approximation 
  
    
      
        
          P
          
            J
          
        
        [
        f
        ]
        (
        x
        )
      
    
    {\displaystyle P_{J}[f](x)}
  
 onto the subspace 
  
    
      
        
          V
          
            k
          
        
      
    
    {\displaystyle V_{k}}
  
, that is, with sampling rate of 2k per unit interval. The difference to the first approximation is given by
 where the difference or detail signals are computed from the detail coefficients as
 with 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
  
 denoting the mother wavelet of the wavelet transform.
 Given the coefficient sequence 
  
    
      
        
          s
          
            (
            M
            )
          
        
      
    
    {\displaystyle s^{(M)}}
  
 for some M < J and all the difference sequences 
  
    
      
        
          d
          
            (
            k
            )
          
        
      
    
    {\displaystyle d^{(k)}}
  
, k = M,...,J − 1, one computes recursively
 for k = J − 1,J − 2,...,M and all 
  
    
      
        n
        ∈
        
          Z
        
      
    
    {\displaystyle n\in \mathbb {Z} }
  
. In the Z-transform notation:
 G. Beylkin, R. Coifman, V. Rokhlin, "Fast wavelet transforms and numerical algorithms"  Comm. Pure Appl. Math., 44 (1991) pp. 141–183 doi:10.1002/cpa.3160440202 (This article has been cited over 2400 times.)