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。这些结构有许多令人兴奋的数学和物理应用，涵盖了从数论到信号处理的范围。我们已经在数论中有了一些应用，并将在数论、分析和信号处理中探索更多。本提案的另一部分是研究傅立叶级数的各种收敛问题。傅里叶级数是一种三角级数，其收敛性问题一直是傅里叶分析和应用中的基础问题。找到级数系数的最弱条件使级数一致收敛，也就是找到某种级数一致收敛的最大群，一直是很重要的，我们也做了一些重要的贡献。我们的目标包括在理论和应用方面进行更多的探索。

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.There are many exciting mathematical and physical applications of these constructions that cover the spectrum from Number Theory to Signal Processing. We already have had several applications in Number Theory, and are going to explore more in Number Theory, Analysis, and Signal Processing.Another part of this proposal is to study various convergence problems of Fourier series. Fourier series are a type of trigonometric series and their convergence problems are always fundamental in Fourier Analysis and applications. Finding the weakest condition on the coefficients of the series so that the series is unformly convergent, i.e., finding the largest group of uniformly convergent series of some type, has been always important and we have made some important contributions. Our goals include more adventures in theory and applications.