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Energy blowup for system approximations and Carleson's theorem

The approximation of stable linear time-invariant systems is a central task in many applications. Therefore, it is important to know if a given approximation process is stable and converges for all signals from the signal space or if it is unstable and diverges for certain signals. Further, in the case of divergence, it is interesting to know whether the set of signals with divergent approximation process has some additional property or inherent structure, like containing subsets that are linear, shift invariant, or closed. In this paper we analyze this question. We will discuss connections to Carleson's theorem, and show that the problem of dense lineability of certain signal sets is equivalent to the Riemann hypothesis.