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Sampling Theory, a Renaissance pp 361–399Cite as

Birkhäuser

A Sheaf-Theoretic Perspective on Sampling

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Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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 these results are dependent on the symmetries of the space of samples. 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.

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Notes

  1. 1.

    An open cover of a topological space X is a collection of open sets whose union is X.

  2. 2.

    N must be at least 3 to use an abstract simplicial complex model of the circle. If N is 1 or 2, one must instead use a CW complex. This does not change the analysis presented here.

  3. 3.

    The interested reader should consider [12] for a practical discussion of several of these conditions in dimensions 1 and 2.

  4. 4.

    |A| represents the cardinality of a set A.

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Acknowledgements

This work was partly supported under Federal Contract No. FA9550-09-1-0643. The author also wishes to thank the editor for the invitation to write this chapter. Portions of this chapter appeared in the proceedings of SampTA 2013, published by EURASIP. Finally, the author wishes to thank Isaac Pesenson for insightful comments that greatly improved the readability of this chapter.

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  1. American University, 4400 Massachusetts Ave NW, Washington, DC, 20016, USA

    Michael Robinson

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  1. Michael Robinson
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Correspondence to Michael Robinson .

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  1. School of Engineering and Science, Jacobs University Bremen, Bremen, Bremen, Germany

    Götz E. Pfander

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Robinson, M. (2015). A Sheaf-Theoretic Perspective on Sampling. In: Pfander, G. (eds) Sampling Theory, a Renaissance. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19749-4_10

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