- Generalized function
In

mathematics ,**generalized functions**are objects generalizing the notion of functions. There is more than one recognised theory. Generalized functions are especially useful in makingdiscontinuous function s more likesmooth function s, and (going to extremes) describing physical phenomena such aspoint charge s. They are applied extensively, especially inphysics andengineering .A common feature of some of the approaches is that they build on

operator aspects of everyday, numerical functions. The early history is connected with some ideas onoperational calculus , and more contemporary developments in certain directions are closely related to ideas ofMikio Sato , on what he calls "algebraic analysis". Important influences on the subject have been the technical requirements of theories ofpartial differential equation s, andgroup representation theory.**ome early history**In the mathematics of the

nineteenth century , aspects of generalized function theory appeared, for example in the definition of theGreen's function , in theLaplace transform , and inRiemann 's theory oftrigonometric series , which were not necessarily theFourier series of anintegrable function . These were disconnected aspects ofmathematical analysis , at the time.The intensive use of the Laplace transform in engineering led to the

heuristic use of symbolic methods, calledoperational calculus . Since justifications were given that useddivergent series , these methods had a bad reputation from the point of view ofpure mathematics . They are typical of later application of generalized function methods. An influential book on operational calculus wasOliver Heaviside 's "Electromagnetic Theory" of 1899.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 samealmost everywhere . That means its value at a given point is (in a sense) not its most important feature. Infunctional analysis a clear formulation is given of the "essential" feature of an integrable function, namely the way it defines alinear functional on other functions. This allows a definition ofweak derivative .During the late 1920s and 1930s further steps were taken, basic to future work. The

Dirac delta function was boldly defined byPaul Dirac (an aspect of hisscientific formalism ); this was to treat measures, thought of as densities (such ascharge density ) like honest functions.Sobolev , working inpartial differential equation theory , defined the first adequate theory of generalized functions, from the mathematical point of view, in order to work withweak solution s of PDEs. Others proposing related theories at the time wereSalomon Bochner andKurt Friedrichs . Sobolev's work was further developed in an extended form by L. Schwartz.**Schwartz distributions**The realization of such a concept that was to become accepted as definitive, for many purposes, was the theory of distributions, developed by

Laurent Schwartz . It can be called a principled theory, based on duality theory fortopological vector space s. Its main rival, inapplied mathematics , is to use sequences of smooth approximations (the 'James Lighthill ' explanation), which is more "ad hoc". This now enters the theory asmollifier theory.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 classicalfunction space s, they are not analgebra . For example it is not meaningful to square theDirac delta function . Work of Schwartz from around 1954 showed that this was an intrinsic difficulty.A simple solution of the multiplication problem is dictated by the

path integral formulation ofquantum mechanics .Since this is required to be equivalent to theSchrödinger theory ofquantum mechanics which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of generalized functionsas shown by H. Kleinert and A. Chervyakov. [*cite journal*] The result is equivalent to what can be derived from

title = Rules for integrals over products of distributions from coordinate independence of path integrals

author = H. Kleinert and A. Chervyakov

journal = Europ. Phys. J.

volume = C 19

issue =

pages = 743--747

year = 2001

doi = 10.1007/s100520100600

url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re303/wardepl.pdfdimensional regularization . [*cite journal*]

title = Coordinate Independence of Quantum-Mechanical Path Integrals

author = H. Kleinert and A. Chervyakov

journal = Phys. Lett.

volume = A 269

issue =

pages = 63

year = 2000

doi = 10.1016/S0375-9601(00)00475-8

url = http://www.physik.fu-berlin.de/~kleinert/305/klch2.pdf**Algebras of generalized functions**Several constructions of algebras of generalized functions have been proposed, among others those by Yu.M.ShirokovYu.M.Shirokov. Algebra of one-dimensional generalized functions.Theoretical and Mathematical Physics,

**39**, 291-301 (1978)http://en.wikisource.org/wiki/Algebra_of_generalized_functions_%28Shirokov%29] and those by E. Rosinger, Y. Egorov, and R. Robinsoncite wanted ] . 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.**Non-commutative algebra of generalized functions**The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function $~F=F(x)~$ to its smooth $F\_\{\; m\; smooth\}$ and its singular $F\_\{\; m\; singular\}$ parts. The product of generalized functions $~F~$ and $~G~$ appears as $(1)~~~~~FG~=~F\_\{\; m\; smooth\}~G\_\{\; m\; smooth\}~+~F\_\{\; m\; smooth\}~G\_\{\; m\; singular\}~+F\_\{\; m\; singular\}~G\_\{\; m\; smooth\}~~~$.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, $~delta(x)^2=0~$. 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 anticommuteYu.M.Shirokov. Algebra of one-dimensional generalized functions.Theoretical and Mathematical Physics, {f 39}, 291-301 (1978)http://en.wikisource.org/wiki/Algebra_of_generalized_functions_%28Shirokov%29] . Few applications of the algebra were suggestedcite 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

number=3

pages=321–324

url=http://www.springerlink.com/content/k2164755540243kw/

doi=10.1007/BF01032729] 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] .**Multiplication of distributions**The problem of "multiplication of distributions", a limitation of the Schwartz distribution theory becomes serious for

non-linear problems.Today the most widely used approach to construct suchassociative differential algebra s is based on J.-F. Colombeau's construction: seeColombeau algebra . These arefactor space s:$G\; =\; M\; /\; N$

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**,$s\; =\; \{\; a\_m:mathbb\; N\; omathbb\; R,\; nmapsto\; n^m\; ;~\; minmathbb\; Z\; \}$.Then for any semi normed algebra (E,P), the factor space will be:$G\_s(E,P)=\; frac\{\{\; fin\; E^\{mathbb\; N\}midforall\; pin\; P,exists\; minmathbb\; Z:p(f\_n)=o(n^m)\{\{\; fin\; E^\{mathbb\; N\}midforall\; pin\; P,forall\; minmathbb\; Z:p(f\_n)=o(n^m).$

In particular, for ("E", "P")=(

**C**,|.|) one gets (Colombeau's) generalized complex numbers (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics,very similar to nonstandard numbers).For ("E", "P") = ("C^{∞}"(**R**),{"p_{k}"})(where "p_{k}" is the supremum of all derivatives of order less than or equal to "k" on the ball of radius "k")one gets Colombeau's simplified algebra.**Injection of Schwartz distributions**This algebra "contains" all distributions "T" of " D' " via the injection

:"j"("T") = (φ

_{"n"}∗ "T")_{"n"}+ "N",where ∗ is the

convolution operation, and:φ

_{"n"}("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^{∞}", 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**× "D"(**R**),with a convenientfilter base on "D"(**R**) (functions of vanishing moments up to order "q").**Sheaf structure**If ("E","P") is a (pre-)sheaf of semi normed algebras on some topological space "X", then "G

_{s}"("E","P") will also have this property.This means that the notion ofrestriction will be defined, which allows to define the 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 thesingular support , i.e., roughly speaking, the closure of the set where the generalized function is not a smooth function (for "E=C^{∞}").**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 defineLars Hörmander 's "wave front set " also for generalized functions.This has an especially important application in the analysis of

propagation of singularities.**Other theories**These include: the "convolution quotient" theory of

Jan Mikusinski , based on thefield of fractions ofconvolution algebras that areintegral domain s; and the theories ofhyperfunction s, based (in their initial conception) on boundary values ofanalytic function s, and now making use ofsheaf theory .**Topological groups**Bruhat introduced a class of

test function s, theSchwartz-Bruhat function s as they are now known, on a class oflocally compact group s that goes beyond themanifold s that are the typicalfunction domain s. The applications are mostly innumber theory , particularly toadelic algebraic group s.André Weil rewroteTate's thesis in this language, characterising the zeta distribution on theidele group ; and has also applied it to theexplicit formula of an L-function .**Generalized section**A further way in which the theory has been extended is as

**generalized sections**of a smoothvector bundle . This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that havecompact support . The most developed theory is that ofDe Rham current s, dual todifferential form s. These are homological in nature, in the way that differential forms give rise toDe Rham cohomology . They can be used to formulate a very generalStokes' theorem .**ee also***

rigged Hilbert space

*generalized eigenfunction

*weak solution **References****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.

* J.-F. Colombeau: New Generalized Functions and Multiplication of the Distributions, North Holland, 1983.

* M. Grosser et al.: Geometric theory of generalized functions with applications to general relativity, Kluwer Academic Publishers, 2001.

* 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)*] (also available online [*http://www.physik.fu-berlin.de/~kleinert/b5 here*] ). See Chapter 11 for products of generalized functions.

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