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.