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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© 1992 Springer Basel AG
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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
Print ISBN: 978-3-0348-9695-5
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