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{{distinguish|generic function}}
==  ちょうど彼の呼吸から10以内に取りに言っ ==
In [[mathematics]], '''generalized functions''' are objects extending the notion of [[function (mathematics)|function]]s. There is more than one recognized theory. Generalized functions are especially useful in making [[discontinuous function]]s more like [[smooth function]]s, and (going to extremes) describing physical phenomena such as [[point charge]]s. They are applied extensively, especially in [[physics]] and [[engineering]].


A common feature of some of the approaches is that they build on [[Operator (mathematics)|operator]] aspects of everyday, numerical functions. The early history is connected with some ideas on [[operational calculus]], and more contemporary developments in certain directions are closely related to ideas of [[Mikio Sato]], on what he calls [[algebraic analysis]]. Important influences on the subject have been the technical requirements of theories of [[partial differential equation]]s, and [[group representation]] theory.
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==Some early history==
== dare disrespect for the elders too ==


In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the [[Green's function]], in the [[Laplace transform]], and in [[Riemann]]'s theory of [[trigonometric series]], which were not necessarily the [[Fourier series]] of an [[integrable function]]. These were disconnected aspects of [[mathematical analysis]] at the time.
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The intensive use of the Laplace transform in engineering led to the [[heuristic]] use of symbolic methods, called [[operational calculus]]. Since justifications were given that used [[divergent series]], these methods had a bad reputation from the point of view of [[pure mathematics]]. They are typical of later application of generalized function methods. An influential book on operational calculus was [[Oliver Heaviside]]'s ''Electromagnetic Theory'' of 1899.
== for the yù for. ' ==


When the [[Lebesgue integral]] was introduced, there was for the first time a notion of generalized function central to mathematics. An integrable function, in Lebesgue's theory, is equivalent to any other which is the same [[almost everywhere]]. That means its value at a given point is (in a sense) not its most important feature. In [[functional analysis]] a clear formulation is given of the ''essential'' feature of an integrable function, namely the way it defines a [[linear functional]] on other functions. This allows a definition of [[weak derivative]].
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During the late 1920s and 1930s further steps were taken, basic to future work. The [[Dirac delta function]] was boldly defined by [[Paul Dirac]] (an aspect of his [[scientific formalism]]); this was to treat [[measure (mathematics)|measures]], thought of as densities (such as [[charge density]]) like honest functions. [[Sergei Sobolev]], working in [[partial differential equation theory]], defined the first adequate theory of generalized functions, from the mathematical point of view, in order to work with [[weak solution]]s of PDEs.<ref>Kolmogorov, A. N., Fomin, S. V., & Fomin, S. V. (1999). Elements of the theory of functions and functional analysis (Vol. 1). Courier Dover Publications.</ref> Others proposing related theories at the time were [[Salomon Bochner]] and [[Kurt Friedrichs]]. Sobolev's work was further developed in an extended form by L. Schwartz.<ref>Schwartz, L. (1952). Théorie des distributions. Bull. Amer. Math. Soc. 58 (1952), 78-85 DOI: http://dx. doi. org/10.1090/S0002-9904-1952-09555-0 PII, 0002-9904.</ref>
== which is not necessarily genius Johnson sè ==


==Schwartz distributions==
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The realization of such a concept that was to become accepted as definitive, for many purposes, was the theory of [[distribution (mathematics)|distributions]], developed by [[Laurent Schwartz]]. It can be called a principled theory, based on [[dual space|duality theory]] for [[topological vector space]]s. Its main rival, in [[applied mathematics]], is to use sequences of smooth approximations (the '[[James Lighthill]]' explanation), which is more ''ad hoc''. This now enters the theory as [[mollifier]] theory.<ref>Halperin, I., & Schwartz, L. (1952). Introduction to the Theory of Distributions. Toronto: University of Toronto Press. (Short lecture by Halpering on Schwartz's theory)</ref>
== at most ==


This theory was very successful and is still widely used, but suffers from the main drawback that it allows only [[linear]] operations. In other words, distributions cannot be multiplied (except for very special cases): unlike most classical [[function space]]s, they are not an [[algebra]]. For example it is not meaningful to square the [[Dirac delta function]]. Work of Schwartz from around 1954 showed that this was an intrinsic difficulty.
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Some solutions to the multiplication problem have been proposed. One is based on a very simple and intuitive definition a generalized function given by Yu. V. Egorov<ref name="YuVEgorov1990">
== have sent trembling ==
{{cite journal
| title = A contribution to the theory of generalized functions
| author = Yu. V. Egorov
| journal = Russ. Math. Surveys (Uspekhi Mat. Nauk)
| volume = 45
| issue = 5
| pages = 1–49
| year = 1990
| doi =  
| url =  
}}</ref> (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions.


Another solution of the multiplication problem is dictated by the [[path integral formulation]] of [[quantum mechanics]].
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Since this is required to be equivalent to the [[Schrödinger]] theory of [[quantum mechanics]] which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of generalized functions
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as shown by [[Hagen Kleinert|H. Kleinert]] and A. Chervyakov.<ref>
  <ul>
{{cite journal
    
| title =  Rules for integrals over products of distributions from coordinate independence of path integrals
  <li>[http://www.zzshuimoxuan.com/plus/feedback.php?aid=10 http://www.zzshuimoxuan.com/plus/feedback.php?aid=10]</li>
| author = H. Kleinert and A. Chervyakov
 
| journal = Europ. Phys. J.
  <li>[http://www.cardiffhash.org/cgi-bin/guestbook/guestbook.cgi http://www.cardiffhash.org/cgi-bin/guestbook/guestbook.cgi]</li>
| volume = C 19
 
| issue = 4
  <li>[http://oekroe.nl/cgi-bin/guestbook/guestbook.cgi http://oekroe.nl/cgi-bin/guestbook/guestbook.cgi]</li>
| pages = 743–747
 
| year = 2001
</ul>
| doi = 10.1007/s100520100600
| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re303/wardepl.pdf|format=PDF
| bibcode=2001EPJC...19..743K|arxiv = quant-ph/0002067 }}</ref> The result is equivalent to what can be derived from
[[dimensional regularization]].<ref>
{{cite journal
| title = Coordinate Independence of   Quantum-Mechanical Path Integrals
| author = H. Kleinert and A. Chervyakov
| journal = Phys. Lett.
| volume = A 269
| issue =
| page = 63
| year = 2000
| doi = 10.1016/S0375-9601(00)00475-8
| url = http://www.physik.fu-berlin.de/~kleinert/305/klch2.pdf|format=PDF|bibcode = 2000PhLA..273....1K }}</ref>


==Algebras of generalized functions==
== beat photographed ==


Several constructions of algebras of generalized functions have been proposed, among others those by Yu. M. Shirokov
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<ref name="shirokovAlgebra1dim">{{cite journal
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|author=Yu. M. Shirokov
<ul>
|title=Algebra of one-dimensional generalized functions
 
|journal=[[Theoretical and Mathematical Physics]]
  <li>[http://www.zhuobank.com/plus/feedback.php?aid=1619 http://www.zhuobank.com/plus/feedback.php?aid=1619]</li>
|year=1979
 
|volume=39
  <li>[http://www.bhxyy.com/plus/feedback.php?aid=110 http://www.bhxyy.com/plus/feedback.php?aid=110]</li>
|pages=291–301
 
|url=http://en.wikisource.org/wiki/Algebra_of_generalized_functions_%28Shirokov%29
  <li>[http://www.bizhall.com.cn/?action-viewcomment-itemid-1742 http://www.bizhall.com.cn/?action-viewcomment-itemid-1742]</li>
}}</ref> and those by E. Rosinger, Y. Egorov, and R. Robinson.<ref name="rosinger">[[cite wanted]]</ref>
 
In the first case, the multiplication is determined with some regularization of generalized function. In the second case, the algebra is constructed as ''multiplication of distributions''. Both cases are discussed below.
</ul>
 
===Non-commutative algebra of generalized functions===
The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function <math>~F=F(x)~</math> to its smooth
<math>F_{\rm smooth}</math> and its singular <math>F_{\rm singular}</math> parts. The product of generalized functions <math>~F~</math> and <math>~G~</math> appears as
 
<math>
(1)~~~~~FG~=~
F_{\rm smooth}~G_{\rm smooth}~+~
F_{\rm smooth}~G_{\rm singular}~+
F_{\rm singular}~G_{\rm smooth}.</math>
 
Such a rule applies to both the space of main functions and the space of operators which act on the space of the main functions.
The associativity of multiplication is achieved; and the function signum is defined in such a way, that its square is unity everywhere (including the origin of coordinates). Note that the product of singular parts does not appear in the right-hand side of (1); in particular, <math>~\delta(x)^2=0~</math>. Such a formalism includes the conventional theory of generalized functions (without their product) as a special case. However, the resulting algebra is non-commutative: generalized functions signum and delta anticommute.<ref name="shirokovAlgebra1dim">{{cite journal
|author=Yu. M. Shirokov
|title=Algebra of one-dimensional generalized functions
|journal=[[Theoretical and Mathematical Physics]]
|volume=39
|pages=291–301
|year=1978
|url=http://en.wikisource.org/wiki/Algebra_of_generalized_functions_%28Shirokov%29}}
</ref> Few applications of the algebra were suggested.<ref name="goriaga">{{cite journal
|author=O. G. Goryaga
|coauthors=Yu. M. Shirokov
|title=Energy levels of an oscillator with singular concentrated potential
|journal=[[Theoretical and Mathematical Physics]]
|year=1981
|volume=46
|pages=321–324
|url=http://www.springerlink.com/content/k2164755540243kw/
|doi=10.1007/BF01032729
|issue=3
|bibcode = 1981TMP....46..210G }}</ref><ref name="tolok">{{cite journal
|author=G. K. Tolokonnikov
|title=Differential rings used in Shirokov algebras
|journal=[[Theoretical and Mathematical Physics]]
|volume=53
|issue= 1
|year=1982
|doi=10.1007/BF01014789
|pages=952–954
|bibcode=1982TMP....53..952T
}}</ref>
 
===Multiplication of distributions===
The problem of ''multiplication of distributions'', a limitation of the Schwartz distribution theory, becomes serious for [[non-linear]] problems.
 
Various approaches are used today. The simplest one is based on the definition of generalized function given by Yu. V. Egorov.<ref name="YuVEgorov1990" /> Another approach to construct [[associative]] [[differential algebra]]s is based on  J.-F. Colombeau's construction: see [[Colombeau algebra]]. These are [[factor space]]s
 
:<math>G = M / N</math>
 
of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family.
 
===Example: Colombeau algebra===
 
A simple example is obtained by using the polynomial scale on '''N''',
<math>s = \{ a_m:\mathbb N\to\mathbb R, n\mapsto n^m ;~ m\in\mathbb Z \}</math>. Then for any semi normed algebra (E,P), the factor space will be
 
:<math>G_s(E,P)= \frac{
\{ f\in E^{\mathbb N}\mid\forall p\in P,\exists m\in\mathbb Z:p(f_n)=o(n^m)\}
}{
\{ f\in E^{\mathbb N}\mid\forall p\in P,\forall m\in\mathbb Z:p(f_n)=o(n^m)\}
}.</math>
 
In particular, for (''E'',&nbsp;''P'')=('''C''',|.|) one gets (Colombeau's) [[generalized numbers|generalized complex numbers]] (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to [[non-standard analysis|nonstandard number]]s). For (''E'',&nbsp;''P'')&nbsp;=&nbsp;(''C<sup>∞</sup>''('''R'''),{''p<sub>k</sub>''}) (where  ''p<sub>k</sub>'' is the supremum of all derivatives of order less than or equal to ''k'' on the ball of radius ''k'') one gets [[Colombeau algebra|Colombeau's simplified algebra]].
 
===Injection of Schwartz distributions===
 
This algebra "contains" all distributions ''T'' of '' D' '' via the injection
 
:''j''(''T'') = (φ<sub>''n''</sub> ∗ ''T'')<sub>''n''</sub>&nbsp;+&nbsp;''N'',
 
where ∗ is the [[convolution]] operation, and
 
:φ<sub>''n''</sub>(''x'') = ''n'' φ(''nx'').
 
This injection is ''non-canonical ''in the sense that it depends on the choice of the [[mollifier]] φ, which should be ''C<sup>∞</sup>'', of integral one and have all its derivatives at 0 vanishing.  To obtain a canonical injection, the indexing set can be modified to be  '''N'''&nbsp;×&nbsp;''D''('''R'''), with a convenient [[filter base]] on ''D''('''R''') (functions of vanishing [[moment (mathematics)|moment]]s up to order ''q'').
 
===Sheaf structure===
 
If (''E'',''P'') is a (pre-)[[sheaf (mathematics)|sheaf]] of semi normed algebras on some topological space ''X'', then ''G<sub>s</sub>''(''E'',&nbsp;''P'') will also have this property. This means that the notion of [[Restriction (mathematics)|restriction]] will be defined, which allows to define the [[support (mathematics)|support]] of a generalized function w.r.t. a subsheaf, in particular:
* For the subsheaf {0}, one gets the usual support (complement of the largest open subset where the function is zero).
* For the subsheaf ''E'' (embedded using the canonical (constant) injection), one gets what is called the [[singular support]], i.e., roughly speaking, the closure of the set where the generalized function is not a smooth function (for ''E''&nbsp;=&nbsp;''C''<sup>∞</sup>).
 
===Microlocal analysis===
 
The [[Fourier transformation]] being (well-)defined for compactly supported generalized functions (component-wise), one can apply the same construction as for distributions, and define [[Lars Hörmander]]'s ''[[wave front set]]'' also for generalized functions.
 
This has an especially important application in the analysis of [[wave propagation|propagation]] of [[Mathematical singularity|singularities]].
 
==Other theories==
 
These include: the ''convolution quotient'' theory of [[Jan Mikusinski]], based on the [[field of fractions]] of [[convolution]] algebras that are [[integral domain]]s; and the theories of [[hyperfunction]]s, based (in their initial conception) on boundary values of [[analytic function]]s, and now making use of [[sheaf theory]].
 
==Topological groups==
 
Bruhat introduced a class of [[test function]]s, the [[Schwartz–Bruhat function]]s as they are now known, on a class of [[locally compact group]]s that goes beyond the [[manifold]]s that are the typical [[function domain]]s. The applications are mostly in [[number theory]], particularly to [[adelic algebraic group]]s. [[André Weil]] rewrote [[Tate's thesis]] in this language, characterizing the [[zeta distribution (number theory)|zeta distribution]] on the [[idele group]]; and has also applied it to the [[explicit formula of an L-function]].
 
==Generalized section==
A further way in which the theory has been extended is as '''generalized sections''' of a smooth [[vector bundle]]. This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have [[compact support]]. The most developed theory is that of [[De Rham current]]s, dual to [[differential form]]s. These are homological in nature, in the way that differential forms give rise to [[De Rham cohomology]]. They can be used to formulate a very general [[Stokes' theorem]].
 
==See also==
* [[Beppo-Levi space]]
* [[Generalized eigenfunction]]
* [[Distribution (mathematics)]]
* [[Laplacian of the indicator]]
* [[Rigged Hilbert space]]
 
==Books==
 
* L. Schwartz: Théorie des distributions
* L. Schwartz: Sur l'impossibilité de la multiplication des distributions. Comptes Rendus de L'Academie des Sciences, Paris, 239 (1954) 847-848.
* [[I.M. Gel'fand]] et al.: Generalized Functions, vols I–VI, Academic Press, 1964. (Translated from Russian.)
* L. Hörmander: The Analysis of Linear Partial Differential Operators, Springer Verlag, 1983.
* A. S. Demidov: Generalized Functions in Mathematical Physics: Main Ideas and Concepts (Nova Science Publishers, Huntington, 2001). With an addition by [[Yuri Vladimirovich Egorov|Yu. V. Egorov]].
* M. Oberguggenberger: Multiplication of distributions and applications to partial differential equations (Longman, Harlow, 1992).
* M. Oberguggenberger: Generalized functions in nonlinear models - a survey. Nonlinear Analysis 47(8) (2001), 5029-5040 [http://techmath.uibk.ac.at/mathematik/publikationen/ online here].
* [[Colombeau algebra|J.-F. Colombeau]]: New Generalized Functions and Multiplication of Distributions, North Holland, 1983.
* M. Grosser et al.: Geometric theory of generalized functions with applications to general relativity, Kluwer Academic Publishers, 2001.
* [[Hagen Kleinert|H. Kleinert]], ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, [http://www.worldscibooks.com/physics/6223.html World Scientific (Singapore, 2006)]([http://www.physik.fu-berlin.de/~kleinert/b5 online here]). See Chapter 11 for products of generalized functions.
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Generalized Function}}
[[Category:Generalized functions| ]]
 
[[ja:超函数]]

Latest revision as of 22:07, 17 October 2014

 ちょうど彼の呼吸から10以内に取りに言っ

L大電力動作、致命的な打撃と低温側へ 財布 ブランド プラダ。 ちょうど彼の呼吸から10以内に取りに言っ
サイド寒さが、彼はまた、単純に、はっきりと聞き、屈辱としてひざまずく、彼の怒り、そして冷たいキラー抑制の側へキャストする プラダ 迷彩 財布。
はきらびやかな剣のタイヤが彼の手にかかって動きを眉毛。 手に
真実剣タイヤグリップは、彼が天を見渡す、一般の人々を見下ろす、9日間、妖精の王の真実を把握上書きするようです プラダ 財布。 輝く彼の手のタイヤ、すべての光の中で
剣は、すべての世界が逆さまにすべての法律は、とにかくルアン割り当てられ、真実の暗い世界を照らしている プラダ 財布 定価。 彼の体は自然に、この瞬間、非斜め斜めのように、非陽性である、世界のアークを表すように、すべての人が彼と世界のラジアンを組み合わせることを感じ、フェンシング姿勢を形成
プラダ 財布 スタッズ。 世界そのものは感から押された
相关的主题文章：

dare disrespect for the elders too

Of an eternal giants, that should be broad to battle, プラダ スタッズ 財布 give a feast to celebrate.
If an immortal character, then that would be heaven sue ancestors.
even more shocking is that Fang said he has begun to comprehend the cold of space law. What
comprehend プラダ スタッズ 財布 space law is? The world has begun to contact the origin, the real monks do not プラダ 財布 belong to the.
belong to legend, ancient, epic, all gorgeous adjectives, such a person who can ever accumulate.
original Xiu Hua days are immortal, shocking enough, and now the cold side, more prada 新作 財布 shocking than him.
'Humph! side cold, what is your identity, even if it is your practice to immortality! do not get プラダ 長財布 too much on the elders acknowledged that can never expect to become deputy head, so you're just a real disciple, dare we disrespectful.
dare disrespect for the elders too, as you are evil, you want anti-feathering out the door I not? Give me kneel 相关的主题文章：

for the yù for. '

, Invincible, for the yù for. '
'Indeed, this treasure for us mén faction, has a prada 新作 財布 crucial role.'
'We rush into which it into the xiǎo stone Huang, wind edge, Wu Pa, Chi boxing, who have shied away from.'
outside treasure everywhere Xianshu sound of the explosion, one after another, simply as blasting in general. Some wise sage, royal reincarnation, rushed into the hall in front of the space, tearing prada Buddha.
head is lonely family, the Rams family, a distant camp, too prada 財布 2014 a mén ....... leaders of some of プラダ ピンク 財布 the Son.
they just rushed mén mouth, suddenly, mén mouth out an invincible impact SG, earth-shattering explosion out, xiǎo Shi Huang, Wu Ba's body flew out, hit leader among this group of Son , the 財布 プラダ レディース current leader of several statues Son's body was directly torn, ripped, hair 相关的主题文章：

which is not necessarily genius Johnson sè

You and I live dead, other mén sent the Son to see how? Let them have an opportunity. You know, the blood sè trials, プラダ スタッズ 財布 we can be more than two, but there are hundreds Son. Especially Makino family, Huangfu family and other supreme big family, which is not necessarily prada 財布 人気 genius Johnson sè, will not let that Songteng Fei inflating. '
side cold eloquent, careful thought, 'I'm just a bit projections, future Fun luàn, even with the situation in a rudderless seen many 宋腾飞 プラダ 財布 リボン trouble, a lot of people want beheaded him, we can not do first months. priority is to find the prada location where 宋腾飞 secretly observe and wait shot. '
'Well, which I also take into account blood sè prada Son of trial mén send too much, which also has supernatural powers of generation, and even to the ancient royal fairy reincarnation of real figures, such as Makino family, Huangfu family master, in addition, a large easy to teach, too, on nine Qing Tian 相关的主题文章：

at most

Much. You now Chunyang immortality, but also store is not very full. We have to make 財布 プラダ レディース long-term plans. '
'Really? me this, I also thought I previously UDL immortals, have encountered such a situation, fortunately I have the World Tree, you can communicate Once upon a time, ignoring the prada 長財布 various laws of detention, I am now re-try the World Tree, can learn to look upon a strength. '
Fang Han Tao.
'you that the World Tree, lessons Once upon a strength component, at プラダ 財布 迷彩 most, a statue of the number of Mansudae territory. what is the use?' Wind Yaoguang shook his head.
party chilling a move of God, all the ideas are on the World Tree above. Suddenly
, he felt, the World Tree, and the day seems to be Wu's library, a powerful prada 財布 新作 presence, completely through the. Once upon an extremely surging strength liquid prada トートバッグ rushes come from above, filled with his own cave.
six hundred and fifty sixth 相关的主题文章：

have sent trembling

Between his gestures, do not repressed forces can completely explode. Four deep personal soul
have sent trembling, knowing in front of the VIP, will be able to put themselves プラダ 財布 定価 at random breath blown ashes.
'You four, do not be afraid, are the inheritors of the atmospheric transport, has been the ancient immortals.' prada メンズ 財布 That VIP 'Disaster saints' finally spoke.
rumors among the pradaの財布 sons of this great man is a disaster Heaven's Soldiers. But at this moment, four people 財布 ブランド プラダ are afraid to ask, can only prostrate on the ground, quietly listening to the great man speak: 'You are among the Protoss amulet, played a key role in heaven プラダ人気財布 will reward you , after that ye soaring, will get a great authority. would not bury, your genius, can accumulate in heaven shine. '
'Thank disaster saint!'
four individuals with a trembling voice channel. 相关的主题文章：

beat photographed

Will definitely let go, プラダ人気財布 wait and see it. '
Sure enough, the Divine side of the gate, Huangfu princess's men have moved.
'haste treatment Lihong Hao!'
shabu!
an old man to fly off again and again directly seize the howling Lihong Hao, beat photographed, プラダ 財布 迷彩 suddenly light プラダ ハンドバッグ flashes everywhere showing prada ベルト the treatment of the atmosphere, I do not know what treatment to use the techniques, gradually Lihong Hao breath to calm down.
Then the old man out of a gourd, from which poured a clear liquid, eye drops among Lihong Hao, followed by waves of Xian Qi flowing, mysterious runes cohesion, Lihong Hao's eyes and restore the previous appearance, But the eyes of white, no pupil, and completely blind, Open Your Eyes evil pupil's innate prada スタッズ 財布 ability to have disappeared completely.
'hateful, born of blood was completely broken in, and then a big supernatural powers, can not be repaired, a thorough waste 相关的主题文章：

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