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.

主要研究员：Sergei K. SUSLOV，提案 ID# DMS-9803443 提案标题：基本傅里叶级数及其扩展 研究项目摘要 特殊函数领域，特别是 q 级数领域，在过去二十年中取得了重大进展。一项重大事件是 Askey-Wilson 多项式的发现。最近在与 q 级数相关的分析、代数和组合学中也存在各种问题。在当前的项目中，我们计划研究基本的傅立叶级数及其扩展。这是分析研究的一个相当新的领域。傅里叶级数和傅里叶-贝塞尔级数有着丰富而深刻的理论。但直到最近，伊斯梅尔、马森和苏斯洛夫才为基本贝塞尔函数建立了连续正交性，并考虑了傅立叶-贝塞尔级数的基本扩展。 Bustoz 和 Suslov 介绍了基本的傅里叶级数，并建立了有关这些级数收敛的几个事实。 Askey提出，伊斯梅尔、马森和苏斯洛夫发现的“贝塞尔型正交性”具有通用性，可以推广到更大一类的基本超几何级数。阿斯基的猜想最近被苏斯洛夫证明。在这个项目中，我们建议发展一种基本傅里叶级数及其更高扩展的理论，类似于傅里叶级数和傅里叶-贝塞尔级数的经典理论。该理论将包括对新 q 正交函数的性质的详细研究、相应级数收敛性的研究以及相关主题。这自然包括某些计算问题：相应的 Sturm-Liouville 问题的特征值只能通过数值方式找到，应该研究这些新级数的收敛性。基本傅里叶级数的明确示例自然会产生一类以前从未从分析和数值角度研究过的新公式。基本傅立叶级数方法可用于研究 q-热方程的解以及数学物理方程的一些其他基本版本。 傅立叶级数的研究在数学领域有着悠久而辉煌的历史。历史上，傅里叶级数是为了求解热方程而引入的，从那时起，这些级数就经常用于各种应用问题。许多现代实分析，包括勒贝格积分的基本理论，都起源于傅里叶级数中的一些深层收敛问题。 如今，人们对傅里叶级数的基本或 q 扩展及其理论产生了很大的兴趣。在这个项目中，我们打算为这项研究奠定良好的基础。我们介绍基本的傅里叶级数，研究它们的主要性质，并考虑数学物理中的一些应用。