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.