A FREQUENCY-RESPONSE FUNCTION FOR LINEAR, TIME-VARYING SYSTEMS

It is well known that the steady-state response of a linear, time-inva riant, finite-dimensional, exponentially stable system to a periodic i nput 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 equ al to the phase of the value of the transfer function at the given fre quency. For this reason the transfer function is also referred to as t he ''frequency response function.'' We present an analogue of this ide a for linear, finite-dimensional time-varying systems, in both the con tinuous- and discrete-time settings. The problem of constructing a tim e-varying system which associates a given output signal to each comple x exponential input signal in a prescribed set can be posed as a model ing question. This leads to a new modeling interpretation for some of the time-varying interpolation problems which have recently been studi ed in the literature and a new motivation for the study of point evalu ation for triangular operators recently introduced by Alpay, Dewilde, and Dym and by the authors for the continuous-time case.