Generalized Functions Theory and Technique

Kanwal, Ram P.

doi:10.1007/978-1-4684-0035-9

Cited by 71 publications

(90 citation statements)

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“…(Refer to [27][28][29] for mathematical treatises on generalized functions and distributions.) In particular, the derivatives do not in general satisfy the Leibniz rule for the derivation of the product, although they satisfy it in an "integrated sense", according to the rule of integration by parts.…”

Section: Hadamard Regularizationmentioning

confidence: 99%

Lorentzian regularization and the problem of point-like particles in general relativity

Blanchet

Faye

2001

Journal of Mathematical Physics

160

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The two purposes of the paper are (1) to present a regularization of the selffield of point-like particles, based on Hadamard's concept of "partie finie", that permits in principle to maintain the Lorentz covariance of a relativistic field theory, (2) to use this regularization for defining a model of stress-energy tensor that describes point-particles in post-Newtonian expansions (e.g. 3PN) of general relativity. We consider specifically the case of a system of two point-particles. We first perform a Lorentz transformation of the system's variables which carries one of the particles to its rest frame, next implement the Hadamard regularization within that frame, and finally come back to the original variables with the help of the inverse Lorentz transformation. The Lorentzian regularization is defined in this way up to any order in the relativistic parameter 1/c 2 . Following a previous work of ours, we then construct the delta-pseudo-functions associated with this regularization. Using an action principle, we derive the stress-energy tensor, made of delta-pseudo-functions, of point-like particles. The equations of motion take the same form as the geodesic equations of test particles on a fixed background, but the role of the background is now played by the regularized metric.

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Section: Hadamard Regularizationmentioning

confidence: 99%

Lorentzian regularization and the problem of point-like particles in general relativity

Blanchet

Faye

2001

Journal of Mathematical Physics

160

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“…The formula for the product of a smooth function p and a derivative of the Dirac delta function in one variable is very simple, namely [16] …”

Section: Basic Formulasmentioning

confidence: 99%

“…Products of this kind are found very frequently in many areas, such as the distributional solution of differential equations [11], [15], [16], and in many of the applications of distribution theory, particularly in Mathematical Physics 1 [2], [3], [10], [12], [13], [17], [18]. For example, the known formulas for the derivatives of power potentials [5], [6] often need to be multiplied by polynomials in the computations performed in the Physics literature [10], [14].…”

Section: Introduction and Notationmentioning

confidence: 99%

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Products of Harmonic Polynomials and Delta Functions

Estrada¹

2018

AAN

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We prove that the product p (x) ∇ 2m δ (x) of a homogeneous polynomial of degree j in n variables, p, and the iterated Laplacian of the Dirac delta function is of the form p (∇) q (∇) δ (x) for some polynomial q if and only if p (x) = |x| 2l p k (x) for some harmonic homogeneous polynomial of some degree k, j = 2l + k. We also show that the product p (x) ∇ 2m δ (x) , where p is homogeneous of degree j, vanishes for all m > j if and only if p is a harmonic polynomial.

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“…See also [1,2]. Here we study the Fourier series ∞ n=−∞ a n e inθ0 by analysing the behavior as ε → 0 of the series ∞ n=−∞ a n e inθ0 φ(nε) for φ ∈ S. Notice that we use the standard notation concerning spaces of distributions and test functions [7,10].…”

Section: Introductionmentioning

confidence: 99%