System approximation with general measurement functionals

The approximation of linear time-invariant (LTI) systems by sampling series is an important topic in signal processing. Recently, it was conjectured [1] and proved [2] that, for every sampling pattern that is a complete interpolating sequence, there exists a universal stable LTI system such that for every oversampling factor there exists a bandlimited input signal such that the approximation process, which is used to approximate the output signal of the LTI system, diverges. This instability of the approximation process shows a fundamental limit of sampling-based signals processing. However, as is shown in this paper, by using more general measurement functionals this divergence can be overcome.