Uniform and Exponential Asymptotics with Applications to Orthogonal Polynomials

DMS-9970489在这项提案中，将开展一些项目。作为研究的主要领域之一，Dunster较早提出的关于转折点和极点的一致渐近理论将被用于研究非经典的正交多项式(重点是Meixner、Pollaczek和Charlier多项式)。这些早先的渐近理论也将用于研究种群生物学中出现的某些奇摄动边值问题。在这个建议中，Dunster还打算将某些经典的一致渐近艾里函数展开式(涉及一个大参数和一个转折点)求和为收敛表示。新的结果将对高阶Bessel函数和高次Legendre函数有直接的应用。所提出问题的一个中心和统一部分是显式和实际误差界的发展，更一般地，这个建议是基于所谓的特殊函数的近似。特殊函数是物理、工程和数学中许多领域中出现的某些方程的解。这是一个非常经典的数学分支，它已经对我们理解物理世界做出了很大贡献。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是非常重要的。此外，在过去的十年里，有一系列新的和令人兴奋的想法导致了渐近方法的精度的显著提高，例如由数学物理学家迈克尔·贝里首创的所谓的“指数渐近”。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。该建议的一个方面是获得对比以前获得的更大范围的变量有效的近似值。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，并且具有重要的理论意义。

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.