有理函数逼近与分段多项式逼近的差异有多大？
How different is rational approximation from piecewise polynomial approximation?
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中文总结 AI 辅助
本文研究有理函数逼近与分段多项式逼近的差异，发现二者在逼近带点奇点的单变量函数时收敛速率相似，但有理逼近常数更小，多变量情形下差异更显著。
中文摘要 AI 辅助
本文的首要目的是证明标题中提出的问题具有研究价值。实际上，对于某些函数类，有理函数逼近与分段多项式逼近的相似程度令人惊讶。在这两种方法中，多项式被研究和应用得更为广泛。其中一个两种方法都得到充分理解且实用价值相当的场景，是逼近具有点奇点的单变量函数。在该场景下，我们可以完整地解答上述问题。我们回顾了该主题的经典文献，这些文献表明两种方法确实能达到相似的收敛速率，但有理逼近的常数要小得多。得益于近年来实用有理逼近的进展，我们可以通过比较实现最优速率的现代数值技术来丰富该讨论。最后，我们通过数值计算表明，在多变量情形下，这种差异会更加显著。
英文摘要
The first aim of this paper is to show that there is merit to the question posed in the title. Indeed, for certain function classes, approximation by rational functions and by piecewise polynomials are surprisingly similar. Between these two, polynomials are more widely studied and more widely used. One setting in which both approaches are equally well understood, and equally practical in use, is that of approximating univariate functions with point singularities. In this context we can fully address the question. We review classical literature on the topic which shows that both approaches do indeed achieve similar convergence rates. However, rational approximations come with significantly smaller constants. Owing to recent advances in practical rational approximation, we can augment the discussion with a comparison of modern numerical techniques that achieve the optimal rates. We end by showing numerically that the difference becomes even more pronounced in several variables.
