The techniques of mathematical approximation theory are applied to the weighted Chebyshev approximation of general transfer functions, as well as loss and phase characteristics. Various methods are shown for implementing the Remez algorithm for rational approximants, and extensions of the existing approximation theory are provided for functionals of rational functions. As a result, explicit expressions are given for the approximation of the modulus of a transfer function or of loss, both with prescribed or arbitrary attenuation poles. Formulas are also included for the phase and phase-delay. These design techniques have been successfully applied to the synthesis of equalizers and simulators for complex systems. A design example illustrates the application of the process to the design of a video cable equalizer.