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.

中文翻译：

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红多尼亚曲线作为单位圆盘上光谱插值的采样轨迹

Rhodonea 曲线是单位圆盘中的经典平面曲线，具有玫瑰的特征形状。在这项工作中，我们使用这些玫瑰曲线作为采样轨迹，为磁盘上的光谱插值创建新节点。通过使用奇偶修正的 Chebyshev-Fourier 基生成插值空间，我们将证明在 rhodonea 节点上插值的不解性。插值方案的连续性、收敛性和数值条件等属性取决于插值空间的谱结构。对于矩形谱索引集，我们表明插值在中心是连续的，勒贝格常数仅以对数方式增长，如果插值函数平滑，则该方案收敛速度很快。最后，