Basic Fourier Series and Their Extensions

基本傅立叶级数及其扩展

• 批准号: 9803443
• 负责人: Sergei Suslov
• 金额: $ 14.05万
• 依托单位: Arizona State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803443&HistoricalAwards=false
• 关键词: Basic; Fourier; Series; Their; Extensions

PRINCIPAL INVESTIGATOR: Sergei K. SUSLOV, PROPOSAL ID# DMS-9803443 PROPOSAL TITLE: Basic Fourier Series and Their Extensions ABSTRACT OF THE RESEARCH PROJECT The area of special functions, and q-series in particular, has seen significant advances in the last twenty years. One major event is the discovery of the Askey-Wilson polynomials. There are also a variety of recent problems in analysis, algebra, and combinatorics related to q-series. In the current project we plan to investigate basic Fourier series and their extensions. This is quite a new area of research in analysis. The Fourier and Fourier-Bessel series have a rich and deep theory. But only recently Ismail, Masson and Suslov have established a continuous orthogonality property for the basic Bessel functions and considered basic extension of the Fourier-Bessel series. Bustoz and Suslov have introduced basic Fourier series and established several facts about convergence of these series. Askey suggested that the "Bessel-type orthogonality" found by Ismail, Masson, and Suslov has a general character and can be extended to a larger class of basic hypergeometric series. Askey's conjecture has recently been proven by Suslov. In this project we propose to develop a theory of basic Fourier series and their higher extensions which is similar to the classical theory of Fourier and Fourier-Bessel series. This theory will include detailed study of properties of the new q-orthogonal functions, investigation of convergence of the corresponding series and related topics. This naturally includes certain computational problems: eigenvalues of the corresponding Sturm-Liouville problem can be found only numerically, investigation of convergence of these new series should be done. Explicit examples of basic Fourier series naturally lead to a new class of formulas never investigated before from the analytical and numerical viewpoint. The method of basic Fourier series can be applied to study solu tions of a q-heat equation and for some other basic versions of the equations of mathematical physics. The study of Fourier series has a long and distiguished history in mathematics. Historically, Fourier series were introduced in order to solve the heat equation, and since then these series have been frequently used in various applied problems. Much of modern real analysis including Lebesgue's fundamental theory of integration had its origin in some deep convergence questions in Fourier series. There is a great deal of interest these days in basic or q-extensions of Fourier series and their theory. In this project we intend to lay a sound foundation for this study. We introduce basic Fourier series, investigate their main properties, and consider some applications in mathematical physics.

首席研究员：Sergei K. Suslov，提案ID＃DMS-9803443提案标题：基本傅立叶系列及其扩展研究摘要研究项目的特殊功能领域，尤其是Q系列，在过去的二十年中取得了重大进展。一个主要事件是发现Askey-Wilson多项式。在分析，代数和与Q系列有关的分析，代数和组合学方面，还有许多最近的问题。在当前的项目中，我们计划研究基本的傅立叶系列及其扩展。这是一个非常新的分析研究领域。傅立叶和傅立叶贝塞尔系列具有丰富而深刻的理论。但是直到最近，Ismail，Masson和Suslov才建立了基本贝塞尔功能的连续正交性，并被认为是傅立叶贝塞尔系列的基本扩展。 Bustoz和Suslov推出了基本的Fourier系列，并建立了有关这些系列融合的几个事实。 Askey建议Ismail，Masson和Suslov发现的“ Bessel型正交性”具有一般特征，可以扩展到更大类的基本超几何序列。阿斯基的猜想最近已被苏斯洛夫证明。在这个项目中，我们建议开发一个基本傅立叶系列的理论及其更高的扩展，这与傅立叶和傅立叶贝塞尔系列的经典理论相似。该理论将包括对新Q-正交功能的性质的详细研究，相应序列和相关主题的收敛性研究。这自然包含某些计算问题：相应的Sturm-liouville问题的特征值只能在数值上找到，应对这些新系列的收敛性进行研究。基本傅立叶系列的明确例子自然会导致从分析和数值观点从未研究过的新一类公式。基本傅立叶系列的方法可以应用于研究Q-Heat方程的解决方案以及一些其他基本版本的数学物理方程。 傅立叶系列的研究在数学方面具有悠久而蒸馏的历史。从历史上看，引入了傅立叶系列以解决热方程，从那时起，这些系列经常在各种应用问题中使用。包括Lebesgue的基本整合理论在内的许多现代真实分析都起源于傅立叶系列中的一些深层融合问题。 如今，对于傅立叶系列的基本或Q延伸及其理论引起了极大的兴趣。在这个项目中，我们打算为这项研究奠定合理的基础。我们介绍基本的傅立叶系列，研究其主要特性，并考虑数学物理学中的某些应用。