A frequency response function for linear, time-varying systems

It is well known that the steady-state response of a linear, time-invariant, finite-dimensional, exponentially stable system to a periodic input signal results, after a phase shift, in a periodic output signal of the same period with amplitude equal to the rescaling of the input amplitude by the modulus of the value of the transfer function at the given frequency. Moreover, the phase shift of the output signal is equal to the phase of the value of the transfer function at the given frequency. For this reason the transfer function is also referred to as the ''frequency response function.'' We present an analogue of this idea for linear, finite-dimensional time-varying systems, in both the continuous- and discrete-time settings. The problem of constructing a time-varying system which associates a given output signal to each complex exponential input signal in a prescribed set can be posed as a modeling question. This leads to a new modeling interpretation for some of the time-varying interpolation problems which have recently been studied in the literature and a new motivation for the study of point evaluation for triangular operators recently introduced by Alpay, Dewilde, and Dym and by the authors for the continuous-time case.