Inmathematics,generalized functions are objects extending the notion offunctions on real or complex numbers. There is more than one recognized theory, for example the theory ofdistributions. Generalized functions are especially useful for treatingdiscontinuous functions more likesmooth functions, and describing discrete physical phenomena such aspoint charges. They are applied extensively, especially inphysics andengineering. Important motivations have been the technical requirements of theories ofpartial differential equations andgroup representations.

A common feature of some of the approaches is that they build onoperator aspects of everyday, numerical functions. The early history is connected with some ideas onoperational calculus, and some contemporary developments are closely related toMikio Sato'salgebraic analysis.

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 theheuristic use of symbolic methods, calledoperational calculus. Since justifications were given that useddivergent series, these methods were questionable 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'sElectromagnetic Theory of 1899.

When theLebesgue 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 each point is (in a sense) not its most important feature. Infunctional analysis a clear formulation is given of theessential 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 basic steps were taken. TheDirac delta function was boldly defined byPaul Dirac (an aspect of hisscientific formalism); this was to treatmeasures, thought of as densities (such ascharge density) like genuine functions.Sergei Sobolev, working inpartial differential equation theory, defined the first rigorous theory of generalized functions in order to defineweak solutions of partial differential equations (i.e. solutions which are generalized functions, but may not be ordinary functions).^[1] Others proposing related theories at the time wereSalomon Bochner andKurt Friedrichs. Sobolev's work was extended byLaurent Schwartz.^[2]

The most definitive development was the theory ofdistributions developed byLaurent Schwartz, systematically working out the principle ofduality fortopological vector spaces. Its main rival inapplied mathematics ismollifier theory, which uses sequences of smooth approximations (the 'James Lighthill' explanation).^[3]

This theory was very successful and is still widely used, but suffers from the main drawback that distributions cannot usually be multiplied: unlike most classicalfunction spaces, they do not form analgebra. For example, it is meaningless to square theDirac delta function. Work of Schwartz from around 1954 showed this to be an intrinsic difficulty.

Some solutions to the multiplication problem have been proposed. One is based on a simple definition of by Yu. V. Egorov^[4] (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions.

Another solution allowing multiplication is suggested by thepath 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 byH. Kleinert and A. Chervyakov.^[5] The result is equivalent to what can be derived fromdimensional regularization.^[6]

Several constructions of algebras of generalized functions have been proposed, among others those by Yu. M. Shirokov^[7] and those by E. Rosinger, Y. Egorov, and R. Robinson.^{[citation needed]}In the first case, the multiplication is determined with some regularization of generalized function. In the second case, the algebra is constructed asmultiplication of distributions. Both cases are discussed below.

The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function${\displaystyle F=F(x)}$ to its smooth${\displaystyle F_{\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}}}$ and its singular${\displaystyle F_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}}}$ parts. The product of generalized functions${\displaystyle F}$ and${\displaystyle G}$ appears as

${\displaystyle FG=F_{\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}}{G}_{\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}}+F_{\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}}{G}_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}}+F_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}}{G}_{\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}}.}$

1

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,${\displaystyle \delta (x{)}^2=0}$. 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 anticommute.^[7] Few applications of the algebra were suggested.^[8][9]

The problem ofmultiplication of distributions, a limitation of the Schwartz distribution theory, becomes serious fornon-linear problems.

Various approaches are used today. The simplest one is based on the definition of generalized function given by Yu. V. Egorov.^[4] Another approach to constructassociativedifferential algebras is based on J.-F. Colombeau's construction: seeColombeau algebra. These arefactor spaces

${\displaystyle G=M/N}$

of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family.

A simple example is obtained by using the polynomial scale onN,${\displaystyle s=\{a_m:\mathbb{N}\to \mathbb{R},n\mapsto n^m;m\in \mathbb{Z}\}}$. Then for any semi normed algebra (E,P), the factor space will be

${\displaystyle G_s(E,P)=\frac{\{f\in E^{\mathbb{N}}\mid \mathrm{\forall }p\in P,\mathrm{\exists }m\in \mathbb{Z}:p(f_n)=o(n^m)\}}{\{f\in E^{\mathbb{N}}\mid \mathrm{\forall }p\in P,\mathrm{\forall }m\in \mathbb{Z}:p(f_n)=o(n^m)\}}.}$

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 tononstandard numbers). For (E, P) = (C^∞(R),{p_k}) (wherep_k is the supremum of all derivatives of order less than or equal tok on the ball of radiusk) one getsColombeau's simplified algebra.

This algebra "contains" all distributionsT of D' via the injection

j(T) = (φ_n ∗T)_n + N,

where ∗ is theconvolution operation, and

φ_n(x) =n φ(nx).

This injection isnon-canonicalin the sense that it depends on the choice of themollifier φ, which should beC^∞, of integral one and have all its derivatives at 0 vanishing. To obtain a canonical injection, the indexing set can be modified to beN × D(R), with a convenientfilter base onD(R) (functions of vanishingmoments up to orderq).

If (E,P) is a (pre-)sheaf of semi normed algebras on some topological spaceX, thenG_s(E, P) will also have this property. This means that the notion ofrestriction will be defined, which allows to define thesupport 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 subsheafE (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 (forE = C^∞).

TheFourier transformation being (well-)defined for compactly supported generalized functions (component-wise), one can apply the same construction as for distributions, and defineLars Hörmander'swave front set also for generalized functions.

This has an especially important application in the analysis ofpropagation ofsingularities.

These include: theconvolution quotient theory ofJan Mikusinski, based on thefield of fractions ofconvolution algebras that areintegral domains; and the theories ofhyperfunctions, based (in their initial conception) on boundary values ofanalytic functions, and now making use ofsheaf theory.

Bruhat introduced a class oftest functions, theSchwartz–Bruhat functions, on a class oflocally compact groups that goes beyond themanifolds that are the typicalfunction domains. The applications are mostly innumber theory, particularly toadelic algebraic groups.André Weil rewroteTate's thesis in this language, characterizing thezeta distribution on theidele group; and has also applied it to theexplicit formula of an L-function.

A further way in which the theory has been extended is asgeneralized 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 currents, dual todifferential forms. 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.

• Beppo-Levi space
• Dirac delta function
• Generalized eigenfunction
• Distribution (mathematics)
• Hyperfunction
• Laplacian of the indicator
• Rigged Hilbert space
• Limit of a distribution
• Generalized space
• Ultradistribution

• Schwartz, L. (1950).Théorie des distributions. Vol. 1. Paris: Hermann.OCLC 889264730. Vol. 2.OCLC 889391733
• Beurling, A. (1961).On quasianalyticity and general distributions (multigraphed lectures). Summer Institute, Stanford University.OCLC 679033904.
• Gelʹfand, Izrailʹ Moiseevič; Vilenkin, Naum Jakovlevič (1964).Generalized Functions. Vol. I–VI. Academic Press.OCLC 728079644.
• Hörmander, L. (2015) [1990].The Analysis of Linear Partial Differential Operators (2nd ed.). Springer.ISBN 978-3-642-61497-2.
• H. Komatsu, Introduction to the theory of distributions, Second edition, Iwanami Shoten, Tokyo, 1983.
• Colombeau, J.-F. (2000) [1983].New Generalized Functions and Multiplication of Distributions. Elsevier.ISBN 978-0-08-087195-0.
• Vladimirov, V.S.; Drozhzhinov, Yu. N.; Zav’yalov, B.I. (2012) [1988].Tauberian theorems for generalized functions. Springer.ISBN 978-94-009-2831-2.
• Oberguggenberger, M. (1992).Multiplication of distributions and applications to partial differential equations. Longman.ISBN 978-0-582-08733-0.OCLC 682138968.
• Morimoto, M. (1993).An introduction to Sato's hyperfunctions. American Mathematical Society.ISBN 978-0-8218-8767-7.
• Demidov, A.S. (2001).Generalized Functions in Mathematical Physics: Main Ideas and Concepts. Nova Science.ISBN 9781560729051.
• Grosser, M.; Kunzinger, M.; Oberguggenberger, Michael; Steinbauer, R. (2013) [2001].Geometric theory of generalized functions with applications to general relativity. Springer.ISBN 978-94-015-9845-3.
• Estrada, R.; Kanwal, R. (2012).A distributional approach to asymptotics. Theory and applications (2nd ed.). Birkhäuser Boston.ISBN 978-0-8176-8130-2.
• Vladimirov, V.S. (2002).Methods of the theory of generalized functions. Taylor & Francis.ISBN 978-0-415-27356-5.
• Kleinert, H. (2009).Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (5th ed.). World Scientific.ISBN 9789814273572. (online hereArchived 2008-06-15 at theWayback Machine). See Chapter 11 for products of generalized functions.
• Pilipovi, S.; Stankovic, B.; Vindas, J. (2012).Asymptotic behavior of generalized functions. World Scientific.ISBN 9789814366847.

• Generalized functions

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