This section provides efficient implementations of specific kinds of linear transformations from to , and from to . These include the discrete cosine transform (DCT), the discrete wavelet transform (DWT), and the discrete Fourier transform (DFT). Whereas an arbitrary linear transformation can be applied by a matrix-vector multiplication in time , these transformations can be applied in time , or even in time . Furthermore, no space is needed to represent the transformation matrix. This efficiency is achieved by exploiting the additional mathematical structure available in these transforms.