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Hadamard regularization

Blanchet, Luc and Faye, Guillaume (2000) Hadamard regularization. Journal of Mathematical Physics, 41 (11). pp. 7675-7714. ISSN 0022-2488. doi:10.1063/1.1308506.

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Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g., 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. We review the concepts of (i) Hadamard "partie finie" of such functions at the location of singular points, (ii) the partie finie of their divergent integral. We present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie (Pf) pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. We introduce and analyze a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. This operator is uniquely defined up to an arbitrary numerical constant. Time derivatives and partial derivatives with respect to the singular points are also investigated. In the course of the paper, all the formulas needed in the application to the physical problem are derived.

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Additional Information:©2000 American Institute of Physics. Received 10 April 2000; accepted 7 July 2000. The authors are grateful to Antoine Sellier for discussion and his interesting comments. This work was supported in part by the National Science Foundation under Grant No. PHY-9900776.
Subject Keywords:general relativity; functions
Issue or Number:11
Record Number:CaltechAUTHORS:BLAjmp00
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3726
Deposited By: Tony Diaz
Deposited On:30 Jun 2006
Last Modified:08 Nov 2021 20:11

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