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); Bagarello JMAA(95)-a0904 [1D, and function], JMAA(02)-a0904 [1D, and quantum field theory]; Steinbauer & Vickers CQG(06)gq; Gsponer EJP(09)mp/06 [intro]; Droz-Vincent JMP(08); Bagarello JMAA(08)-a0904 [any dimension, and function]; Skakala a0908 [for tensorial distributions]; > s.a. Colombeau Algebra.
@ Properties: Smirnov TMP(07)mp/05 [localization properties].
@ Ultradistributions: Bollini et al TMP(99), Bollini & Rocca TMP(04), TMP(04), et al IJTP(07)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)sep [electromagnetism, dipoles]; Gsponer EJP(07) [spherical symmetry, electrodynamics]; > s.a. diffeomorphisms; particle models [pointlike electron]; quantum field theory techniques [operator-valued distributions]; solutions in general relativity with matter [distributional sources]; types of metrics [distributional curvature].

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 references: Aguirregabiria et al AJP(02)feb [converging sequences]; Boykin AJP(03)may [sequences converging to '(x)]; Jackson AJP(08)-a0708 [attribution]; Towers JCP(09) [discretized via finite-difference methods]; Galapon JPA(09) [converging sequences].
@ Uses: Blinder AJP(03)aug [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].

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