  Sampling theory has traditionally drawn tools from functional and complex
analysis. Past successes, such as the Shannon-Nyquist theorem and recent
advances in frame theory, have relied heavily on the application of geometry
and analysis. The reliance on geometry and analysis means that the results are
geometrically rigid. There is a subtle interplay between the topology of the
domain of the functions being sampled, and the class of functions themselves.
Bandlimited functions are somewhat limiting; often one wishes to sample from
other classes of functions. The correct topological tool for modeling all of
these situations is the sheaf; a tool which allows local structure and
consistency to derive global inferences. This chapter develops a general
sampling theory for sheaves using the language of exact sequences, recovering
the Shannon-Nyquist theorem as a special case. It presents sheaf-theoretic
approach by solving several different sampling problems involving
non-bandlimited functions. The solution to these problems show that the
topology of the domain has a varying level of importance depending on the class
of functions and the specific sampling question being studied.
