Blanchet, Luc and Faye, Guillaume (2000) Hadamard regularization. Journal of Mathematical Physics, 41 (11). pp. 76757714. ISSN 00222488. http://resolver.caltech.edu/CaltechAUTHORS:BLAjmp00

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Abstract
Motivated by the problem of the dynamics of pointparticles in high postNewtonian (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 powerlike 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 partiefinie (Pf) pseudofunction. The multiplication of pseudofunctions is defined by the ordinary (pointwise) product. We construct a deltapseudofunction 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 pseudofunctions, 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.
Item Type:  Article 

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. PHY9900776. 
Subject Keywords:  general relativity; functions 
Record Number:  CaltechAUTHORS:BLAjmp00 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:BLAjmp00 
Alternative URL:  http://dx.doi.org/10.1063/1.1308506 
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3726 
Collection:  CaltechAUTHORS 
Deposited By:  Tony Diaz 
Deposited On:  30 Jun 2006 
Last Modified:  26 Dec 2012 08:55 
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