This section provides efficient implementations of specific kinds of linear transformations in ''RR^n'' and ''CC^n''. 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 ''O(n^2)'', these transformations can be applied in time ''O(n log(n))'', or even in time ''O(n)''. Furthermore, no space is needed to represent the transformation matrix. This efficiency is achieved by exploiting the additional mathematical structure available in these transforms.