Research Interests
My research is generally in the area
of operator theory. My recent work involved the determination
of the spectra of variable-coefficient Toeplitz matrices. These matrices
can be viewed as a discrete analogue of one-dimensional pseudo-differential
operators or a class of generalized Toeplitz matrices. I have, with
my collaborator, used the ideas from operator theory to obtain new results
on asymptotics of determinants of these matrices, thus extended
the classical theory of Toeplitz eterminants. One such is the variable-coefficient
version of the strong Szego-Widom limit theorem.
More recent work involves with the
asymptotics for the singular values of certain generalized Toeplitz matrices,
and Toeplitz minors. These new results obtained are the extension of
Tracy-Widom's formula on Toeplitz minors, and Widom's second order
asymptotic formula for the singular values of the usual Toeplitz matrices.
My other interests include a general
method of Object-Oriented Analysis and Design in C++ programming, and problems
in numerical analysis. Specifically, I am interested in problems
on low and high dimensional spline fittings and stability questions in
systems of stiff differential equations.
Papers
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A Trace Formula for Variable-Coefficient Toeplitz
Matrices with Symbols of Bounded Variation. Journal
of Mathematical Analysis and Applications 222, (1998) 505-504.
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[with Torsten Ehrhardt] Asymptotic Behavior
of Variable-Coefficient Toeplitz Determinants. Journal of
Fourier Analysis and Applications.. Vol 7, Issue 1, (2001) 71 - 92.
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A Trace Formula for a class of Block
Variable-Coefficient Toeplitz Matrices. Journal of Integral Equations
and Operator Theory. 45 (2003) 359 - 374
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On
the Singular Values of Generalized Toeplitz Matrices. Journal
of Integral Equations and Operator Theory. 49 (2004) 239 - 254
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On the Limit of Some Generalized Toeplitz-Like
Determinants. Preprint
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Asymptotic Behavior of the Singular Values
of Generalized Toeplitz Matrices with Piecewise Continuous Symbols.(in
preparation)
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On the Determinants of Generalized Toeplitz
+ Hankel matrices. (in preparation)
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A Trace Formula for the Product of Toeplitz
Operators. (in preparation)