Like the discrete Fourier transform, the discrete wavelet transform (DWT) is a linear operation that defines a forward and inverse relationship between the time-domain and the frequency-domain, also called the wavelet domain. This relationship is expressed through the use of basis functions. In the case of the DFT, trigonometric sines and cosines of varying angles are used. In the case of the DWT, the basis functions are more complicated and usually called mother functions or wavelets. Also like the DFT, the DWT is orthogonal, making many operations computationally efficient. For example, the inverse wavelet transform, when viewed as a matrix operator, is simply the transpose of the forward transform.
Most of the usefulness of wavelets relies on the fact that wavelet transforms can usefully be severely truncated-that is, they can be effectively turned into sparse expressions. This property is a result of the simultaneous compact representation of the wavelet basis functions in the time and frequency domains. See WTN for an example using the wavelet transform.