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Sampling on Unions of Shifted Lattices in One Dimension

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Book cover Harmonic Analysis and Applications

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

Abstract

We give a complete solution to the problem of sampling and interpolation of functions in PW σ(R) on finite unions of shifted lattices in R of the form \( \Lambda j = \frac{1} {{2\sigma _j }}Z + \alpha _j ,j = 1, \ldots ,m \) where σ j > 0, α j ∈ R, and ∑ j σ j =σ. At points where more than one lattice intersect, we sample the function and its derivatives. None of the results or techniques employed is new, but a systematic and elementary treatment of this situation does not seem to exist in the literature. Sampling on unions of shifted lattices includes classical sampling, bunched or periodic sampling, and sampling with derivatives. Such sampling sets arise in deconvolution, tomography, and in the theory of functions bandlimited to convex regions in the plane.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, The Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Bjarte Rom

  2. Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, 22030, USA

    David Walnut

Authors
  1. Bjarte Rom
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  2. David Walnut
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Editor information

Editors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Christopher Heil

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© 2006 Birkhäuser Boston

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Rom, B., Walnut, D. (2006). Sampling on Unions of Shifted Lattices in One Dimension. In: Heil, C. (eds) Harmonic Analysis and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4504-7_13

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