勒贝格常数与多变量多项式插值的勒贝格常数
Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous-Chebyshev nodes
AI总结：
研究多变量多项式插值在利萨如-切比雪夫节点上的绝对条件数，推导勒贝格常数的上下界，基于多项式插值与傅里叶级数的多面体部分和的勒贝格常数关系，通过多面体边长对数的乘积确定界大小。
摘要：
为分析多变量多项式插值在利萨如-切比雪夫节点上的绝对条件数，我们推导了相应的勒贝格常数的上下界。证明基于勒贝格常数与多面体部分和的傅里叶级数勒贝格常数之间的关系。所获界的大小由所考虑多面体集合边长的对数乘积决定，并表现出与切比雪夫网格多项式插值勒贝格常数大小相同的特性。
英文摘要：
To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous-Chebyshev node points, we derive upper and lower bounds for the respective Lebesgue constant. The proof is based on a relation between the Lebesgue constant for the polynomial interpolation problem and the Lebesgue constant linked to the polyhedral partial sums of Fourier series. The magnitude of the obtained bounds is determined by a product of logarithms of the side lengths of the considered polyhedral sets and shows the same behavior as the magnitude of the Lebesgue constant for polynomial interpolation on the tensor product Chebyshev grid.
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