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dc.contributor.authorFulling, Stephen A. ( )
dc.contributor.authorGustafson, Robert ( )
dc.date.accessioned2019-11-21T18:57:47Z
dc.date.available2019-11-21T18:57:47Z
dc.date.issued1999-03-01
dc.identifier.citationFulling, S. A., & Gustafson, R. A. (1999). Some properties of Riesz means and spectral expansions. Electronic Journal of Differential Equations, 1999(06), pp. 1-39.en_US
dc.identifier.issn1072-6691
dc.identifier.urihttps://digital.library.txstate.edu/handle/10877/8856
dc.description.abstractIt is well known that short-time expansions of heat kernels correlate to formal high-frequency expansions of spectral densities. It is also well known that the latter expansions are generally not literally true beyond the first term. However, the terms in the heat-kernel expansion correspond rigorously to quantities called Riesz means of the spectral expansion, which damp out oscillations in the spectral density at high frequencies by dint of performing an average over the density at all lower frequencies. In general, a change of variables leads to new Riesz means that contain different information from the old ones. In particular, for the standard second-order elliptic operators, Riesz means with respect to the square root of the spectral parameter correspond to terms in the asymptotics of elliptic and hyperbolic Green functions associated with the operator, and these quantities contain ``nonlocal'' information not contained in the usual Riesz means and their correlates in the heat kernel. Here the relationship between these two sets of Riesz means is worked out in detail; this involves just classical one-dimensional analysis and calculation, with no substantive input from spectral theory or quantum field theory. This work provides a general framework for calculations that are often carried out piecemeal (and without precise understanding of their rigorous meaning) in the physics literature.en_US
dc.formatText
dc.format.extent39 pages
dc.format.medium1 file (.pdf)
dc.language.isoen_USen_US
dc.publisherTexas State University, Department of Mathematicsen_US
dc.sourceElectronic Journal of Differential Equations, 1999, San Marcos, Texas: Southwest Texas State University and University of North Texas.
dc.subjectRiesz meansen_US
dc.subjectSpectral asymptoticsen_US
dc.subjectHeat kernelen_US
dc.titleSome Properties of Riesz Means and Spectral Expansionsen_US
txstate.documenttypeArticle
dc.rights.licenseCreative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/


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