Uniform and Exponential Asymptotics with Applications to Orthogonal Polynomials

一致和指数渐进及其在正交多项式中的应用

• 批准号: 9970489
• 负责人: Timothy Dunster
• 金额: $ 4.56万
• 依托单位: San Diego State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2001-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970489&HistoricalAwards=false
• 关键词: Uniform; Exponential; Asymptotics; Applications; Orthogonal

DMS-9970489In this proposal a number of projects will be undertaken. As oneof the main areas of research, uniform asymptotic theories involving acoalescing turning point and pole, developed earlier by Dunster, will beused in the study of non-classical orthogonal polynomials (with an emphasison the Meixner, Pollaczek and Charlier polynomials). These earlierasymptotic theories will also be used in the study of certain singularlyperturbed boundary value problems arising from population biology. Also inthis proposal Dunster intends to sum certain classical uniform asymptoticAiry function expansions (involving a large parameter and a turning point)into convergent representations. The new results would then have animmediate application to Bessel functions of large order and Legendrefunctions of large degree. A central and unifying component of the proposedproblems is the development of explicit and realistic error bounds.More generally, this proposal is based on the approximation of theso-called special functions. Special functions are solutions of certainequations which appear in many areas of physics, engineering andmathematics. This is a very classical branch of mathematics that hasalready contributed much to our understanding of the physical world.Numerical and theoretical results concerning special functions are still ofgreat importance in virtually all of the physical sciences. Morever, duringthe past decade there have been a series of new and exciting ideas leadingto significant improvements in the accuracy of asymptotic methods, such asthe so-called "exponential asymptotics" pioneered by the mathematicalphysicist Michael Berry. In this project Dunster intends to use and extendthese ideas in a number of directions. One aspect of the proposal is toobtain approximations which are valid for a wider range of variables thanobtained previously. An immediate advantage of these more powerfulapproximations is that there is greater flexibility in their use innumerical applications, as well as having important theoreticalimplications.

DMS-9970489在该提案中将进行许多项目。 作为研究的主要领域，涉及邓斯特（Dunster）早期开发的涉及涉及交流的转折点和极点的统一渐近理论将在非古典正交多项式的研究中使用（强调Meixner，Pollaczek，Pollaczek和Charlier polynomials）。 这些早期的症状理论也将用于研究人口生物学引起的某些单层状边界价值问题。 此外，提案邓斯特（Dunster）也打算将某些经典均匀渐近函数函数扩展（涉及大参数和转折点）汇总成收敛表示。 然后，新的结果将使对大订单和大范围的贝塞尔功能的应用程序进行动画应用。所提出的问题的一个中心和统一的组成部分是开发明确和现实的误差。通常，该建议基于该指称的特殊功能的近似。特殊功能是某些方程式的解决方案，这些解决方案出现在许多物理，工程和数学领域。 这是数学的一个非常古典的分支，已经对我们对物理世界的理解做出了很大的贡献。关于特殊功能的数字和理论结果在几乎所有物理科学中仍然至关重要。此外，在过去的十年中，已经有一系列新的令人兴奋的想法，导致了渐近方法准确性的重大改善，例如由数学物理学家迈克尔·贝里（Michael Berry）开创的所谓的“指数渐近学”。在这个项目中，Dunster打算在许多方向上使用和扩展这些想法。该提案的一个方面是toobtain近似值，这些近似值对于先前已被视为的变量有效。 这些更强大的应用程序的直接优势是，它们的使用无数应用以及具有重要的理论刺激性具有更大的灵活性。