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Sampling Theory with Average Values on the Sierpinski Gasket

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Abstract

In the case of some fractals, sampling with average values on cells is more natural than sampling on points. In this paper we investigate this method of sampling on SG and SG\(_{3}\). In the former, we show that the cell graph approximations have the spectral decimation property and prove an analog of the Shannon sampling theorem. We also investigate the numerical properties of these sampling functions and make conjectures that allow us to look at sampling on infinite blowups of SG. In the case of SG\(_{3}\), we show that the cell graphs have the spectral decimation property but show that it is not useful for proving an analogous sampling theorem.

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Authors and Affiliations

  1. Mathematics Department, Duke University, Durham, NC, 27708, USA

    Robert J. Ravier

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Robert S. Strichartz

Authors
  1. Robert J. Ravier
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  2. Robert S. Strichartz
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Correspondence to Robert J. Ravier.

Additional information

Communicated by Mauro Maggioni.

Robert J. Ravier was supported in part by the National Science Foundation, Grant DMS-0739164. Robert S. Strichartz was supported in part by the National Science Foundation, Grant DMS-1162045.

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Ravier, R.J., Strichartz, R.S. Sampling Theory with Average Values on the Sierpinski Gasket. Constr Approx 44, 159–194 (2016). https://doi.org/10.1007/s00365-016-9341-7

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  • DOI: https://doi.org/10.1007/s00365-016-9341-7

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