One might thing that in a pure (noiseless) generalized inverse problem there is no place for sampling theory at all. Yet sampling theory has managed to work itself into the problem anyway, by a remarkable conceptual feat. If the real world has no sampling distribution, then we shall invent one, not by imbedding our data in a set of possible data, but by imbedding the whole real world in a set of possible worlds (just as Everett (1957) did in quantum theory). The one real, finite time series Y = {y_0,...y_N} that actually exists is regarded as only a “sample” drawn from some hypothetical ensemble of other infinitely long series—and we are back in business!
In http://bayes.wustl.edu/etj/articles/fromhere.pdf pg20, Jaynes (<1985) makes the following comment in a discussion of time-series, which I take as being a bit sarcastic and to MWI’s detriment: