q-Series, Combinatorics and Computer Algebra,
a Joint Summer Research Conference in the Mathematical Sciences, Mount Holyoke College,
South Hadley, Massachusetts, 21-25 June, 1998.

Analysis Seminar

Research interests: Extensions of Euler's summation formulas and the zeta function: latex, dvi, ps, pdf

Current NSF Grant:
http://www.nsf.gov/cgi-bin/showaward?award=9803443

PRINCIPAL INVESTIGATOR: Sergei K. SUSLOV
GRANT ID#: DMS-9803443
PROJECT TITLE: Basic Fourier Series and Their Extensions
PROGRAM: Analysis Program

ABSTRACT OF THE RESEARCH PROJECT

The study of Fourier series has a long and distinguished 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.

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 [1]. 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 the field of
basic hypergeometric functions and classical analysis. The Fourier and Fourier-Bessel
series have a rich and deep theory. See, for example, [2-4] and references therein.
But only recently Ismail, Masson and Suslov [5] have established a continuous orthogonality
property for the basic Bessel functions on a q-quadratic grid and considered basic extension
of the Fourier-Bessel series. At the same time Bustoz and Suslov [6] have introduced basic
Fourier series and established several important facts about convergence of these series.
Richard Askey suggested that the "Bessel type orthogonality" found by Ismail, Masson,
and Suslov in [5] 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 [7, 8].
In this project we propose to develop a theory of basic Fourier series and their higher
extensions which is similar, in some sense, 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,
numerical 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 beauty of just a few known examples makes the study of basic Fourier series
a fascinating problem in the area of classical analysis and special functions
that many researchers in the field will find interesting. The method of basic Fourier series
can be applied to study solutions of a q-heat equation and for some other basic versions of
the equations of mathematical physics.

REFERENCES

1. R. A. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials
    that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. #319 (1985).
2. N. K. Bary, A Treatise on Trigonometric Series, in two volumes, Macmillan,
    New York, 1964.
3. A. Zygmund, Trigonometric Series, second edition, Cambridge University Press,
    Cambridge, 1968.
4. G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition,
    Cambridge University Press, Cambridge, 1944.
5.  M. E. H. Ismail, D. R. Masson, and S. K. Suslov, The q-Bessel functions on
     a q-quadratic grid, to appear in the Proceedings of CRM workshop on algebraic
     methods and q-special functions, Montreal, May 1996.
6.  J. Bustoz and S. K. Suslov, Basic analog of Fourier series on a q-quadratic grid,
     Mathematical Sciences Research Institute Preprint No. 1997-060, Berkeley,
     California, 1997, 42 pp;
     see also http://lefty.msri.org/publications/preprints/online/1997-060.html;
     Methods and Applications of Analysis 5 (1998) #1, 1-38.
7.  S. K. Suslov, Some orthogonal 8_\phi_7-functions, J. Phys. A: Math. Gen. 30
     (1997), 5877-5885.
8.  S. K. Suslov, Some orthogonal very-well-poised 8_\phi_7-functions that
     generalize Askey-Wilson polynomials, Mathematical Sciences Research Institute
     Preprint No. 1997-070, Berkeley, California, 1997, 31 pp;
     see also  http://lefty.msri.org/publications/preprints/online/1997-070.html.