General Principle
Let F be any sampling method, i.e. a linear map from the Hilbert space of square-integrable functions to complex space .
In our example, the vector space of sampled signals is n-dimensional complex space. Any proposed inverse R of F (reconstruction formula, in the lingo) would have to map to some subset of . We could choose this subset arbitrarily, but if we're going to want a reconstruction formula R that is also a linear map, then we have to choose an n-dimensional linear subspace of .
This fact that the dimensions have to agree is related to the Nyquist–Shannon sampling theorem.
The elementary linear algebra approach works here. Let (all entries zero, except for the kth entry, which is a one) or some other basis of . To define an inverse for F, simply choose, for each k, an so that . This uniquely defines the (pseudo-)inverse of F.
Of course, one can choose some reconstruction formula first, then either compute some sampling algorithm from the reconstruction formula, or analyze the behavior of a given sampling algorithm with respect to the given formula.
Read more about this topic: Signal Reconstruction
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