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在本建议书中，将开展多个项目。 邓斯特早期发展的包含收敛转折点和极点的一致渐近理论是研究非经典正交多项式（重点是Meixner多项式、Pollaczek多项式和Charlier多项式）的主要领域之一。 这些早期的渐近理论也将被用于研究某些奇异摄动边值问题所产生的人口生物学。 此外，在这个建议邓斯特打算总结某些经典的一致渐近Airy函数展开（涉及一个大参数和一个转折点）到收敛表示。 新结果将直接应用于高阶贝塞尔函数和高阶勒让德函数。所提出的问题的一个中心和统一的组成部分是明确和现实的误差界的发展。更一般地说，这个建议是基于对这些所谓的特殊函数的近似。特殊函数是物理、工程和数学等许多领域中出现的某些方程的解。 这是数学的一个非常经典的分支，对我们理解物理世界有很大的贡献。关于特殊函数的数值和理论结果在几乎所有的物理科学中仍然非常重要。然而，在过去的十年里，出现了一系列新的令人兴奋的想法，导致渐近方法的准确性有了显著的提高，例如由物理学家迈克尔·贝里开创的所谓的“指数渐近”。在这个项目中，邓斯特打算在许多方向上使用和扩展这些想法。该建议的一个方面是获得近似值，这是有效的一个更广泛的变量比以前获得。 这些更强大的近似的一个直接优点是，在数值应用中使用它们有更大的灵活性，以及具有重要的理论意义。