When an approximant is accurate on the interval, it is only natural to try to extend it to several-dimensional domains. In the present article, we make use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several times differentiable functions to interpolate on two-dimensional starlike domains parametrized in polar coordinates. In radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially toward analytic functions. In circular direction, we deploy linear rational trigonometric barycentric interpolants, which converge similarly rapidly for periodic functions, but now for conformally shifted equispaced nodes. We introduce a variant of a tensor-product interpolant of the above two schemes and prove that it converges exponentially for two-dimensional analytic functions up to a logarithmic factor and with an order limited only by the order of differentiability for real functions, if the boundary is as smooth. Numerical examples confirm that the shifts permit to reach a much higher accuracy with significantly less nodes, a property which is especially important in several dimensions.

翻译：当一个相近的星体在间隔线上准确时,试图将其扩展至多个维域是自然的。在目前的条款中,我们利用线性理性的巴里中心间极极迅速聚集到解析和数倍不同功能,在极坐标上对二维星形域进行内插。在辐射方向中,我们接触相向转移的切比谢夫节点的跨极者,它们向解析函数呈指数性趋同。在圆方向中,我们部署线性三角对称中巴里中心间极者,它们与周期函数的相近,但现在却与正向偏移的天线性节点相交。我们引入了上述两种图案的一个变异的变体,并证明它为两维的解析函数而成指数性趋近于一个对数系数,而且如果边界是平滑的,则仅受不同功能的顺序限制。数字示例证实,这种转移允许以显著的节点到达一个高得多的精确度,在几个方面都特别重要。