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 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.