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 在此提案中将开展许多项目。 作为主要研究领域之一，邓斯特早期提出的涉及合并转折点和极点的一致渐近理论将用于非经典正交多项式（重点是 Meixner、Pollaczek 和 Charlier 多项式）的研究。 这些早期的渐近理论也将用于研究群体生物学中产生的某些奇扰动边值问题。 此外，在这个提案中，邓斯特打算将某些经典的一致渐近艾里函数展开式（涉及大参数和转折点）求和为收敛表示。 新的结果将立即应用于大阶贝塞尔函数和大阶勒让函数。所提出问题的核心和统一组成部分是明确且现实的误差界限的发展。更一般地，该提议基于所谓的特殊函数的近似。特殊函数是物理、工程和数学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，已经为我们理解物理世界做出了很大贡献。有关特殊函数的数值和理论结果在几乎所有物理科学中仍然非常重要。此外，在过去的十年中，出现了一系列令人兴奋的新想法，导致渐近方法的准确性显着提高，例如数学物理学家迈克尔·贝里（Michael Berry）首创的所谓“指数渐近学”。在这个项目中，邓斯特打算在多个方向上使用和扩展这些想法。该提案的一方面是获得对比以前获得的更广泛的变量有效的近似值。 这些更强大的近似的直接优点是它们在数值应用中的使用具有更大的灵活性，并且具有重要的理论意义。