Classical Summability, Classical Banach Spaces, Sequence Spaces, Operators on Banach Spaces.
Current research interests
Operators on Banach Sequences Spaces, Absolutely Summing Operators.
My research is primarily in the areas of classical and functional analytic aspects of summability theory. Summability theory started with the study of divergent series and integrals. It has led to many applications to other areas of mathematics such as Fourier Series, which are used to study periodic phenomena, acceleration of convergence of series and integrals, certain problems concerning approximation theory, and analytic continuation. I am interested, in particular, in seeing how certain concepts that arise in classical summability can be adapted to deal with questions in the theory of Banach Spaces. Banach Spaces are fundamental in functional analysis and have been an important subject of research during most of the twentieth century and will continue to play a central role in the mathematics of this century.
I collaborated with J. Boos who is a professor at the Fern Universitaet in Hagen, Germany in the writing of a book that was published by Oxford University Press in 2000. This book, whose title is "Classical and Modern Methods in Summability," (ISBN 0 19 850165 X) is very broad in scope and covers most areas of current research in summability. No book of this nature has appeared since G. H. Hardy's book "Divergent Series" which was published by Oxford University Press in 1949.
Along with my research into Banach sequence spaces, I am now involved in writing an introductory book on number theory. The subject matter discussed will cover much of the standard material in "elementary number theory" and will be continued to give an introduction to algebraic number theory. The approach to algebraic number theory will be designed to illustrate and motivate many notions encountered by students when they begin their study of abstract algebra.
Telephone: 519-661-3638 x 86523