"Rational approximants, sequences and series, and applications"

“有理近似、数列和级数以及应用”

• 批准号: 222884-2012
• 负责人: Zhou, Ping
• 金额: $ 0.87万
• 依托单位: St. Francis Xavier University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570219
• 关键词: Rational; approximants; sequences; series; applications

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.

理性逼近是逼近理论的主要研究领域之一，其中也包括帕岱勒逼近。多元理性近似的研究相对较新，部分原因是涉及计算困难，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，我们将在数论和傅里叶分析中探索更多。