Distributions| Distributions |
In General
> s.a. [analysis]; functional analysis.
* Idea: A type of
generalized function; The theory was introduced by Dirac and formalized
by Schwarz, motivated by quantum physics.
$ Def: An element of the
dual space of the Schwarz space \(C_0^\infty({\mathbb R}^n)\).
* Schwarz space:
The space of all (test or smearing) functions f :
\(\mathbb R\)n →
\(\mathbb R\) which are infinitely differentiable and fall off fast
enough at infinity (e.g., compact support, or faster than any power);
It is used to define distributions.
@ Books: 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;
Duistermaat & Kolk 10;
El Kinani & Oudadess 10.
@ General references: Estrada & Fulling JPA(02) [defined by singular functions];
in Waldmann a1208-ln [on manifolds].
@ Products: Colombeau 83,
BAMS(90);
Oberguggenberger 92;
Köhler CQG(95);
Bagarello JMAA(95)-a0904 [1D, and δ 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 δ-function];
Skákala a0908,
PhD-a1107 [for tensorial distributions];
Kim JMP(10)
[multiplication and convolution of distributions and ultradistributions];
Nigsch & Sämann a1309
[overview, and applications in general relativity];
> 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];
> s.a. Extrafunctions;
tensor fields [distributional].
@ 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. fractional analysis [fractional derivative];
fourier transform.
* Properties:
It satisfies ∇2
r−1
= −4π δ(r) in 3D.
* Approximations:
the following 1-parameter families of functions approximate
δ(x) as L→∞,
δL(x):= \(\int_{-L/2}^{L/2}\) dk exp{i2πkx} πxL)/(πx) and \((2L)^{-1}{\rm e}^{-|x|/L}\) .
* Result: It can be represented using the non-extensive-statistical-mechanics q-exponential function eqix = [1+(1−q)ix]1/(1−q) as
δ(x) = (2−q)/(2π) \(\int_{-\infty}^{+\infty}\) dk eq−ikx ; here, q ∈ [1, 2] and q = 1 is the usual exponential representation .
@ General references: Jackson AJP(08)-a0708 [attribution];
Towers JCP(09) [discretized via finite-difference methods];
Jáuregui & Tsallis JMP(10)-a1004,
Mamode JMP(10),
Plastino & Rocca JMP(11)-a1012 [and q-exponential function];
Katz & Tall FS(12)-a1206 [19th-century roots];
Kempf et al JPA(14)-a1404 [properties and applications in quantum field theory];
Sicuro & Tsallis PLA(17)-a1705
[generalized representation in d dimensions in terms of q-exponential functions].
@ Converging sequences: Aguirregabiria et al AJP(02)feb; Boykin AJP(03)may [sequences converging to δ'(x)];
Galapon JPA(09).
@ Uses: Blinder AJP(03)aug [re fields of points charges and dipoles];
Bondar et al AJP(11)apr-a1007 [differentiation, and use in quantum mechanics].
@ Generalizations:
Rosas-Ortiz in(06)-a0705 [Dirac-Infeld-Plebański improper delta function];
Ducharme a1403 [complex, and the quantized electromagnetic field];
Zhang a1607 [on vector spaces and matrix spaces].
Finite Part Distribution
> s.a. integration theory [finite-part integration].
$ Def: The finite part of a function
f(x) is the distribution defined by FP \(\int_{-\infty}^\infty\)
φ(x) f(x) dx = non-divergent terms
in the power-series expansion around ε = 0 of
\(\int_{-\infty}^{-\epsilon}\) φ(x) f(x) dx
+ \(\int_\epsilon^\infty\) φ(x) f(x) dx.
* Examples: For the
case of r−2, with
a second-order pole at x = 0, this means
FP \(\int_0^\infty\) r−2 f(r) dr = −f(0) + \(\int_0^1\) r−2 [f(r) − f(0) −rf '(0)] dr + \(\int_1^\infty\) r−2 f(r) dr ;
i.e., expand f(r) in series around the singular point
of φ(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];
Seriu AOT-a1003 [asymptotic principal values];
Galapon JMP(16)-a1512 [Cauchy Principal Value and finite part integral as values of absolutely convergent integrals].
> Online resources:
see Wikipedia pages on the Hadamard finite part and Cauchy principal value.
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