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Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk

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Abstract

Rhodonea curves are classical planar curves in the unit disk with the characteristic shape of a rose. In this work, we use these rose curves as sampling trajectories to create novel nodes for spectral interpolation on the disk. By generating the interpolation spaces with a parity-modified Chebyshev–Fourier basis, we will prove the unisolvence of the interpolation on the rhodonea nodes. Properties such as continuity, convergence, and numerical condition of the interpolation scheme depend on the spectral structure of the interpolation space. For rectangular spectral index sets, we show that the interpolant is continuous at the center, that the Lebesgue constant grows only logarithmically, and that the scheme converges fast if the interpolated function is smooth. Finally, we show that the scheme can be implemented efficiently using a fast Fourier method and that it can be applied to define a Clenshaw–Curtis quadrature on the disk.

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Acknowledgements

I want to thank both referees for their valuable feedback and the constructive comments that helped to improve the quality of this manuscript. This research was partially funded by GNCS-IN\(\delta \)AM and by the European Union Horizon 2020 research and innovation programme ERA-PLANET, Grant Agreement No. 689443 (GEOEssential).

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  1. Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste 63, 35121, Padua, Italy

    Wolfgang Erb

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  1. Wolfgang Erb
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Correspondence to Wolfgang Erb.

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Communicated by Yuan Xu.

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Erb, W. Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk. Constr Approx 53, 281–318 (2021). https://doi.org/10.1007/s00365-019-09495-w

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  • DOI: https://doi.org/10.1007/s00365-019-09495-w

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