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

Abstract

Rhodonea curves are classical planar curves in the unit disk with the characteristic shape of a rose. In this work, we use point samples along such rose curves as node sets for a novel spectral interpolation scheme on the disk. By deriving a discrete orthogonality structure on these rhodonea nodes, we will show that the spectral interpolation problem is unisolvent. The underlying interpolation space is generated by a parity-modified Chebyshev-Fourier basis on the disk. This allows us to compute the spectral interpolant in an efficient way. Properties as continuity, convergence and numerical condition of the 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, the Lebesgue constant grows logarithmically and that the scheme converges fast if the function under consideration is smooth. Finally, we derive a Clenshaw-Curtis quadrature rule using function evaluations at the rhodonea nodes and conduct some numerical experiments to compare different parameters of the scheme.Keywords: Spectral interpolation on the disk, rhodonea curves, intersection and boundary nodes of rhodonea curves, parity-modified Chebyshev-Fourier series, Clenshaw-Curtis quadrature on the disk, numerical condition and convergence of interpolation schemes 2010 MSC: 41A05,42A16,65D05,65T50