Basic Fourier Series and Their Extensions

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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.