"Rational approximants, sequences and series, and applications"

有理逼近，包括Padé逼近，是逼近论的主要研究领域之一.多元有理逼近的研究是相对较新的，部分原因是涉及的计算困难，并适合于计算机代数软件包的实验。我们的部分意图是使尽可能多的研究尽可能系统，并构造尽可能多的显式逼近。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Padé逼近。这个建议的一部分是保持势头，探索更明确的结构。这些构造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几个应用，并将在数论和傅立叶分析中探索更多。 另一部分是研究傅里叶级数的各种收敛问题和某些无穷级数的可和性。傅里叶级数是一类三角级数，其收敛性问题一直是傅里叶分析及其应用中的基本问题。找到级数系数的最弱条件，使级数一致收敛（或可和），即，寻找某种类型的一致收敛级数（或可和因子）的最大群一直是重要的，我们也做出了一些重要的贡献。我们的目标包括在理论和应用方面取得更多进展。

Rational approximation, including Padé approximation, is one of the main areas in Approximation Theory. The study of multivariate rational approximants is relatively new, partly because of the computational 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 and Fourier Analysis. Another part of this proposal is to study various convergence problems of Fourier series and summability properties of some infinite 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 uniformly convergent (or summable), i.e., finding the largest group of uniformly convergent series of some type (or summability factors), has been always important, and we have made some important contributions. Our goals include more advances in theory and applications.