In mathematics, **generalized functions** are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering.

- Some early history
- Schwartz distributions
- Algebras of generalized functions
- Non-commutative algebra of generalized functions
- Multiplication of distributions
- Example: Colombeau algebra
- Injection of Schwartz distributions
- Sheaf structure
- Microlocal analysis
- Other theories
- Topological groups
- Generalized section
- See also
- Books
- References

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 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 equations, and group representation theory.

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.

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.

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.

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 measures, thought of as densities (such as charge density) like genuine 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 solutions of partial differential equations.^{ [1] } Others proposing related theories at the time were Salomon Bochner and Kurt Friedrichs. Sobolev's work was further developed in an extended form by Laurent Schwartz.^{ [2] }

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 for topological vector spaces. 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.^{ [3] }

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 spaces, 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 was an intrinsic difficulty.

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^{ [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 of the multiplication problem is dictated by the path integral formulation of quantum mechanics. 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 as shown by H. Kleinert and A. Chervyakov.^{ [5] } The result is equivalent to what can be derived from dimensional 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 as *multiplication 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 to its smooth and its singular parts. The product of generalized functions and appears as

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, . 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 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.^{ [4] } Another approach to construct associative differential algebras is based on J.-F. Colombeau's construction: see Colombeau algebra. These are factor spaces

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 on **N**, . Then for any semi normed algebra (E,P), the factor space will be

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 ^{∞}*(

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

If (*E*,*P*) is a (pre-)sheaf of semi normed algebras on some topological space *X*, then *G _{s}*(

- 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*=*C*^{∞}).

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 propagation of singularities.

These include: the *convolution quotient* theory of Jan Mikusinski, based on the field of fractions of convolution algebras that are integral domains; and the theories of hyperfunctions, based (in their initial conception) on boundary values of analytic functions, and now making use of sheaf theory.

Bruhat introduced a class of test functions, the Schwartz–Bruhat functions as they are now known, on a class of locally compact groups that goes beyond the manifolds that are the typical function domains. The applications are mostly in number theory, particularly to adelic algebraic groups. André Weil rewrote Tate's thesis in this language, characterizing the zeta distribution on the idele group; and has also applied it to the explicit formula of an L-function.

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 currents, dual to differential forms. 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.

- L. Schwartz: Théorie des distributions
- L. Schwartz: Sur l'impossibilité de la multiplication des distributions. Comptes Rendus de l'Académie des Sciences de 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 Yu. V. Egorov.
- M. Oberguggenberger: Multiplication of distributions and applications to partial differential equations (Longman, Harlow, 1992).
- Oberguggenberger, M. (2001). "Generalized functions in nonlinear models - a survey".
*Nonlinear Analysis*.**47**(8): 5029–5040. doi:10.1016/s0362-546x(01)00614-9. - 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.
- H. Kleinert,
*Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, 4th edition, World Scientific (Singapore, 2006)(online here). See Chapter 11 for products of generalized functions.

In mathematics, a **Lie group** is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

**Distributions**, also known as **Schwartz distributions** or **generalized functions**, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

In mathematics, particularly linear algebra and functional analysis, a **spectral theorem** is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In mathematics, **complex geometry** is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, a **Colombeau algebra** is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this.

In mathematics, the **support** of a real-valued function *f* is the subset of the domain containing the elements which are not mapped to zero. If the domain of *f* is a topological space, the support of *f* is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.

In mathematics, **Hodge theory**, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold *M* using partial differential equations. The key observation is that, given a Riemannian metric on *M*, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called **harmonic**.

In mathematics, **complex multiplication** (**CM**) is the theory of elliptic curves *E* that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties *A* having *enough* endomorphisms in a certain precise sense. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

In mathematics, **Kähler differentials** provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

In geometry, a **Poisson structure** on a smooth manifold is a Lie bracket on the algebra of smooth functions on , subject to the Leibniz rule

In mathematical analysis a **pseudo-differential operator** is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory.

In mathematics, **mollifiers** are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a function which is rather irregular, by convolving it with a mollifier the function gets "mollified", that is, its sharp features are smoothed, while still remaining close to the original nonsmooth (generalized) function.

In mathematics, **hyperfunctions** are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958 in Japanese,, building upon earlier work by Laurent Schwartz, Grothendieck and others.

In mathematical analysis, more precisely in microlocal analysis, the **wave front (set)** WF(*f*) characterizes the singularities of a generalized function *f*, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.

In mathematics, a **Schwartz–Bruhat function**, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A **tempered distribution** is defined as a continuous linear functional on the space of Schwartz–Bruhat functions.

In mathematics, the **Schwartz kernel theorem** is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz have a two-variable theory that includes all reasonable bilinear forms on the space of test functions. The space itself consists of smooth functions of compact support.

A **product integral** is any product-based counterpart of the usual sum-based integral of calculus. The first product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Other examples of product integrals are the geometric integral, the bigeometric integral, and some other integrals of non-Newtonian calculus.

- ↑ Kolmogorov, A. N., Fomin, S. V., & Fomin, S. V. (1999). Elements of the theory of functions and functional analysis (Vol. 1). Courier Dover Publications.
- ↑ Schwartz, L (1952). "Théorie des distributions".
*Bull. Amer. Math. Soc*.**58**: 78–85. doi: 10.1090/S0002-9904-1952-09555-0 . - ↑ Halperin, I., & Schwartz, L. (1952). Introduction to the Theory of Distributions. Toronto: University of Toronto Press. (Short lecture by Halperin on Schwartz's theory)
- 1 2 Yu. V. Egorov (1990). "A contribution to the theory of generalized functions".
*Russian Math. Surveys*.**45**(5): 1–49. Bibcode:1990RuMaS..45....1E. doi:10.1070/rm1990v045n05abeh002683. - ↑ H. Kleinert and A. Chervyakov (2001). "Rules for integrals over products of distributions from coordinate independence of path integrals" (PDF).
*Eur. Phys. J. C*.**19**(4): 743–747. arXiv: quant-ph/0002067 . Bibcode:2001EPJC...19..743K. doi:10.1007/s100520100600. - ↑ H. Kleinert and A. Chervyakov (2000). "Coordinate Independence of Quantum-Mechanical Path Integrals" (PDF).
*Phys. Lett*. A 269 (1–2): 63. arXiv: quant-ph/0003095 . Bibcode:2000PhLA..273....1K. doi:10.1016/S0375-9601(00)00475-8. - 1 2 Yu. M. Shirokov (1979). "Algebra of one-dimensional generalized functions".
*Theoretical and Mathematical Physics*.**39**(3): 291–301. Bibcode:1979TMP....39..471S. doi:10.1007/BF01017992. - ↑ O. G. Goryaga; Yu. M. Shirokov (1981). "Energy levels of an oscillator with singular concentrated potential".
*Theoretical and Mathematical Physics*.**46**(3): 321–324. Bibcode:1981TMP....46..210G. doi:10.1007/BF01032729. - ↑ G. K. Tolokonnikov (1982). "Differential rings used in Shirokov algebras".
*Theoretical and Mathematical Physics*.**53**(1): 952–954. Bibcode:1982TMP....53..952T. doi:10.1007/BF01014789.

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