Multivariate padé approximants, fourier series, and applications

先验近似是对泰勒级数的一种合理推广，已经被研究了一个多世纪。埃尔米特对e的超越性的证明涉及到构造这样的近似值。他的学生帕德帕尔系统地研究了这些近似，并以他的名字命名。多元pad<s:1>近似值的研究相对较新，部分原因在于涉及的计算困难，并且适合于用计算机代数包进行实验。我们的部分目的是使研究尽可能系统化，并尽可能多地构建显式近似值。在过去的几年中，我们成功地构造了一些多元函数的显式多元pad<s:1>近似。这一建议的一部分是保持势头，探索更明确的结构。

Padé approximants are a rational generalization of Taylor series and have been studied for over a century. Hermite's proof of transcendence of e involved constructing such approximants. His student Padé explored the approximants systematically and got his name attached to them. The study of multivariate Padé approximants is relatively new, partly because of the computation difficulties involved, and lends itself to experimentation with computer algebra packages. Part of our intention is to make as much of the study as systematic as possible, and to construct as many of the explicit approximants as we can. In the past few years, we successfully constructed explicit multivariate Padé approximants to some multivariate functions. Part of this proposal is to keep the momentum and explore more explicit constructions.