Distributions| Distributions |
In General > s.a. [analysis]; functional
analysis.
* Idea: A type of generalized
function, introduced by Dirac and formalized by Schwarz.
$ Def: An element of
the dual space of the Schwarz space C0infty(Rn).
* Schwarz space: The
space of all (test or smearing) functions f : Rn
→ R which are infinitely differentiable and fall off
fast enough at infinity (e.g.,
compact support, or faster than any power); Used to define
distributions.
@ General references: Schwartz 50, 51 [original]; Gel'fand & Shilov
64, 68, 67; Gel'fand & Vilenkin
64; Gel'fand, Graev & Vilenkin 66; Lighthill 64; in Roos
69; in Adams 75, ch1; Richards & Youn 90; Estrada & Fulling JPA(02)
[defined by singular functions].
@ Products: Colombeau 83, BAMS(90);
Oberguggenberger 92; Köhler CQG(95);
Gsponer mp/06 [intro];
Droz-Vincent JMP(08).
@ Properties: Smirnov TMP(07)mp/05 [localization
properties].
@ Ultradistributions: Bollini et al TMP(99),
Bollini & Rocca
TMP(04),
TMP(04), et al ht/06 [convolution].
@ Generalizations: Kunzinger & Steinbauer AAM(02)m.FA/01 [for
sections of vector bundles, tensors], TAMS(02)m.FA/01 [pseudo-Riemannian
geometry]; Kunzinger MfM(02)m.FA/01,
et al PLMS(03)m.FA/02 [manifold-valued];
Dragovich ITSF(98)mp/04 [Adelic];
Colombeau mp/07,
a0708[adapted
to non-linear calculations].
@ Applications: Skinner & Weil AJP(89)
[electromagnetism, dipoles]; Gsponer EJP(07)
[spherical symmetry, electrodynamics]; > s.a. diffeomorphisms, quantum
field theory
techniques [operator-valued
distributions], solutions in general relativity
with matter [distributional
sources].
Dirac Delta Function > s.a. fourier
transform.
* Properties: Satisfies 
2 r–1 =
–4
(r)
in 3D.
* Approximations:
L(x):= 
–L/2L/2
dk exp{i2kx}
= sin(xL)/(x)
.
@ General: Aguirregabiria et al AJP(02) [converging sequences]; Boykin AJP(03) [sequences
converging to '(x)].
@ Uses: Blinder AJP(03) [re fields of points charges and dipoles].
@ Generalizations: Rosas-Ortiz in(06)-a0705 [Dirac-Infeld-Plebanski
improper delta function].
Finite Part Distribution
$ Def: The finite part
of a function f(x) is the distribution
defined by FP
–inftyinfty 
(x) f(x)
dx = non-divergent terms in
the power series expansion around 
=
0 of –infty–eps(x) f(x)
dx +
epsinfty(x) f(x)
dx.
* Examples: For the case of r–2,
with a second order pole at x =
0, this means
FP 0infty r–2 f(r)
dr = –f(0)
+ 01 r–2 [f(r) – f(0) – rf '(0)]
dr + 1infty r–2 f(r)
dr ,
i.e., expand f(r) in series around the singular point of (r), and give a prescription for how to integrate those terms which cause
trouble: in
the
above case, the first two.
@ References: in Blanchet & Faye JMP(00)gq,
JMP(01)gq/00 [and
pointlike particles].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
15 jul 2008