We give fiber or planar lightwave circuit (PLC) architectures to implement the discrete Fourier transform (DFT) and the discrete Hartley transform (DHT) directly in the optical domain. In both cases, we present a recursive approach for the decimation-in-frequency algorithm, considering parallel and serial input configurations. We also describe PLC devices for high-speed optical filtering and data compression, based on discrete wavelet transforms (DWT) and discrete cosine transform (DCT).