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Hadamard regularization. (English) Zbl 0986.46024

Summary: Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g.,3PN) approximations of general relativity, the authors consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. They 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.

They present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. They 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. They 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.

MSC:
46F10Operations with distributions (generalized functions)