星形域上的线性重心有理插值
A linear barycentric rational interpolant on starlike domains
摘要：
当一个近似式在区间上是精确的时，很自然地会试图将其扩展到几维域。本文利用线性有理重心插值快速收敛于解析函数和几次可微函数的事实，在极坐标参数化的二维星形区域上进行插值。在径向方向上，我们从事插值共形位移切比雪夫节点，指数收敛到解析函数。在圆周方向上，我们部署线性有理三角重心插值，它收敛同样迅速的周期函数，但现在共形移动等间距节点。我们介绍了一个变种的张量积插值的上述两个计划，并证明它收敛指数为二维解析函数-对数因子-和一个命令仅限于由真正的功能的可微性顺序（提供边界享有相同的顺序的可微性）。数值例子证实，允许的转变，以达到更高的精度与显着更少的节点，属性是特别重要的，在几个维度。
英文摘要：
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 (provided that the boundary enjoys the same order of differentiability). Numerical examples confirm that the shifts permit to reach a much higher accuracy with significantly fewer nodes, a property which is especially important in several dimensions.
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