A sheaf-theoretic perspective on sampling

TL;DR

This paper introduces a sheaf-theoretic framework for sampling theory, extending classical results like Shannon-Nyquist to more general function classes and emphasizing the role of topology in sampling problems.

Contribution

It develops a general sheaf-based sampling theory using exact sequences, unifying classical and new sampling problems beyond bandlimited functions.

Findings

Sheaf theory provides a flexible topological framework for sampling.

Classical Shannon-Nyquist theorem is recovered as a special case.

Topology's importance varies with function class and sampling context.

Abstract

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.