Abstract
A general principle is proposed that in all the usual asymptotic expansions for tr f(T) where f is a “general” function and T is a Toeplitz, Wiener-Hopf, or pseudodifferential operator, each term of the expansion is an integral of (or, more generally, some distribution applied to) f(σ*), where σ* is a “Symbol” associated with that term of the expansion. These Symbols are defined in terms of σ, the symbol of the given operator, but have larger domain. Although nothing so general is proved we consider several operators with smooth or nonsmooth symbol, evaluate early terms of asymptotic expansions associated with them, and show that they can all be cast into the form described.
Research supported by National Science Foundation Grant DMS 8822906.
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References
E. Basor. Trace formulas for Toeplitz matrices with piecewise continuous symbol J. Math. Anal. Applic. 120 (1986).
A. Böttcher and B. Silbermann. Analysis of Toeplitz Operators. Akademic-Verlag, Berlin, 1989.
X. Fu. Asymptotics of Toeplitz matrices with symbols of bounded variation. Ph.D. Thesis, Univ. of California, Santa Cruz, 1991.
I. C. Gohberg and I. A. Feldman. Convolution Equations and Projection Methods for Their Solution. Amer. Math. Soc. Transi. Math. Monographs 41 (1974).
I. Gohberg and N. Krupnik. On the algebra generated by Toeplitz matrices. Funcl. Anal. Appl. 3 (1969).
L. Hörmander. The Analysis of Linear Partial Differential Operators III. Springer-Verlag, Berlin, 1985.
L. M. Libkind. Asymptotics of the eigenvalues of Toeplitz forms. Math. Notes. 11 (1972).
H. Widom. Families of Pseudodifferential Operators. In Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York, 1978.
_____. On a class of integral operators with discontinuous symbol. J. Int. Eqs. Operator Th. 5 (1982).
_____. A trace formula for Wiener-Hopf operators. J. Operator Th. 8 (1982).
_____. Asymptotic Expansion for Pseudodifferential Operators on Bounded Domains. Lecture Notes in Math. 1152, Springer-Verlag, Berlin, 1986.
_____. Asymptotic expansions and stationary phase for operators with discontinuous symbol. To appear.
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Widom, H. (1992). Symbols and Asymptotic Expansions. In: Gohberg, I. (eds) Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations. Operator Theory: Advances and Applications, vol 58. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8596-6_7
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DOI: https://doi.org/10.1007/978-3-0348-8596-6_7
Publisher Name: Birkhäuser, Basel
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