Basic Fourier Series and Their Extensions

主要负责人：谢尔盖·K. SUSLOV，提案ID# DMS-9803443 基本傅立叶级数及其扩展 研究项目摘要 特殊函数领域，特别是q系列， 在过去的二十年里取得了重大进展。一个主要事件是 Askey-Wilson多项式的发现也有 分析、代数和组合学中的各种最新问题 关于Q系列在目前的项目中，我们计划调查 基本傅立叶级数及其扩展这是一个全新的领域 研究分析。傅里叶级数和傅里叶-贝塞尔级数 丰富而深刻的理论。但直到最近伊斯梅尔、马松和苏斯洛夫 已经建立了一个连续的正交性质的基本 Bessel函数和Fourier-Bessel的基本推广 系列. Bustoz和Suslov介绍了基本的傅里叶级数， 建立了几个关于这些级数收敛性的事实。 Askey认为Ismail发现的“贝塞尔型正交性”， Masson和Suslov具有一般性，可以推广到 基本超几何级数的较大类。Askey的猜想 最近被苏斯洛夫证实了在这个项目中，我们建议开发 一种基本傅立叶级数及其高阶扩展的理论， 类似于经典的傅立叶和傅立叶-贝塞尔理论 系列.这一理论将包括详细研究的性质， 新的q-正交函数，收敛性研究， 系列及相关专题。这自然包括 某些计算问题：相应的Sturm-Liouville问题的特征值只能在数值上找到，调查 这些新系列的收敛性应该得到保证。明确实例 基本傅里叶级数的性质自然地导致一类新的公式 以前从未从分析和数值上研究过 观点基本傅里叶级数的方法可以用来研究 一个q-热方程的解和一些其他基本形式的 数学物理方程 傅里叶级数的研究有着悠久而独特的历史， 数学历史上，傅立叶级数被引入，以便 求解热方程，从那时起，这些级数 经常用于各种应用问题。现代真实的 包括勒贝格整合基本理论在内的分析， 它起源于Fourier级数的某些深收敛问题。 现在人们对基本的或q-扩展有很大的兴趣 傅立叶级数及其理论在这个项目中，我们打算奠定 为这项研究奠定了坚实的基础。我们介绍基本的傅里叶级数， 研究它们的主要性质，并考虑一些应用 数学物理学中。

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.