omparison of HTMs. Norms and criteria for comparing HTMs are especially useful for constructing approxima- tions to HTMs. They allow us to quantify the notion of HTMs being "almost equal". Since HTMs represent LPTV operators (systems), the theory on HTM norms is closely related to the theory on operator norms. A.1 Operator norms and the comparison of operators Comparing two (LPTV) operators H1,H2ES;-So, (A.1) where Si is the Banach space [Huts80] containing the input signals and So the one containing the output signals, requires us to define a notion of a distance d(H1, H2) between them ¹. The most natural way to do so, is by comparing the outputs they produce for some relevant set Si of test inputs. For continuous linear operators, this leads to [Huts80] d(H1,H2)=|H|-H2||, (A.2) (A.3) Here, II II is the norm defined over the space of input signals Si and .llo is the norm defined over the space of output signals So. In what follows, we drop the subscripts ; and 0. Which norm to use should be clear from the context. The distance HH-H2|| can now be used to quantify the "degree of equality" of the two operators H1 and H2: stating that H2 is close to H1, or H2 22 H1, implies that one expects HI-H2|| to be small. But what is "small"? The most sensible answer seems to be that, for each input (A.4) 1 Note that Si and So need not necessarily to be the same, e.g. in passing from the continuous-time to the discrete-time domain. In another example, Si is the space of all signals with low-frequency content only, while the signals in So can have both low- and high-frequency content, i.e. Si C So. A.3 EXPRESSING LPTV OPERATOR NORMS IN TERMS OF THE CORRESPONDING HTM ELEMENTS max |H2[u]-H1[u]|| < E , (A.5) UESi |H1[u]|| with E << 1 some suitably chosen threshold level. However, there is still one matter open for debate. What if, for a certain V # 0, H1[v] =0? In that case, (A.4) becomes impossible to satisfy. To overcome this, we need to introduce the notion of a zero signal-level, i.e. there is a threshold level Eollvll for whi