NationMaster - Encyclopedia: Distribution (mathematics)

In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. They extend the concept of derivative to all integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations. They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions are distributions, such as the Dirac delta distribution. A probability distribution describes the values and probabilities that a random event can take place. ... Distribution can mean: Often distribution is the spatial property of being scattered. ... Analysis has its beginnings in the rigorous formulation of calculus. ... This article is about functions in mathematics. ... A probability distribution describes the values and probabilities that a random event can take place. ... For other uses, see Derivative (disambiguation). ... Integrability is a mathematical concept used in different areas. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...

"Generalized functions" were introduced by Sergei Sobolev in 1935. They were independently introduced in the late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions. Sobolev, Sergei Lvovich (Russian: Сергей Львович Соболев) (6 October 1908- 3 January 1989) was a Russian mathematician, working in mathematical analysis and partial differential equations. ... Laurent Schwartz (5 March 1915 â€“ 4 July 2002 in Paris) was a French mathematician. ...

Basic idea

The basic idea is to identify functions with abstract linear functionals on a space of unproblematic test functions (conventional and well-behaved functions). Operators on distributions can be understood by moving them to the test function. In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ...

be a locally integrable function, and let In mathematics, a locally integrable function is a function which is integrable on any compact set. ...

be a smooth (that is, infinitely differentiable) function with compact support (i.e., identically zero outside of some bounded set). The function φ is the "test function." We then set In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... For other uses, see Derivative (disambiguation). ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

This is a real number which linearly and continuously depends on φ. One can therefore think of the function f as a continuous linear functional on the space which consists of all the "test functions" φ. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the term functional is applied to certain functions. ...

Similarly, if P is a probability distribution on the reals and φ is a test function, then

is a real number that continuously and linearly depends on φ: probability distributions can thus also be viewed as continuous linear functionals on the space of test functions. This notion of "continuous linear functional on the space of test functions" is therefore used as the definition of a distribution.

Such distributions may be multiplied with real numbers and can be added together, so they form a real vector space. In general it is not possible to define a multiplication for distributions, but distributions may be multiplied with integrable functions. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

To define the derivative of a distribution, we first consider the case of a differentiable and integrable function f : R → R. If φ is a test function, then we have

using integration by parts (note that φ is zero outside of a bounded set and that therefore no boundary values have to be taken into account). This suggests that if S is a distribution, we should define its derivative S' by In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ...

It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.

Example: The Dirac delta (so-called Dirac delta function) is the distribution defined by The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...

It is the derivative of the Heaviside step function: For any test function $varphi$ , The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

H' = δ

. $varphi(infty)=0$ because of compact support. Similarly, the derivative of the Dirac delta is the distribution

This latter distribution is our first example of a distribution which is neither a function nor a probability distribution.

Formal definition

In the sequel, real-valued distributions on an open subset U of Rⁿ will be formally defined. (With minor modifications, one can also define complex-valued distributions, and one can replace Rⁿ by any smooth manifold.) In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ...

First, the space D(U) of test functions on U needs to be explained. A function φ : U → R is said to have compact support if there exists a compact subset K of U such that φ(x) = 0 for all x in U K. The elements of D(U) are the infinitely differentiable functions φ : U → R with compact support (also known as bump functions). This is a real vector space. We turn it into a topological vector space by stipulating that a sequence (or net) (φ_k) converges to 0 if and only if there exists a compact subset K of U such that all φ_k are identically zero outside K, and for every ε > 0 and natural number d ≥ 0 there exists a natural number k₀ such that for all k ≥ k₀ the absolute value of all d-th derivatives of φ_k is smaller than ε. With this definition, D(U) becomes a complete topological vector space (in fact, a so-called LF-space). In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... For other senses of this word, see sequence (disambiguation). ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, an LF-space is a topological vector space V that is a countable strict inductive limit of FrÃ©chet spaces. ...

The dual space of the topological vector space D(U), consisting of all continuous linear functionals S : D(U) → R, is the space of all distributions on U; it is a vector space and is denoted by D'(U). The dual pairing between a distribution S in D′(U) and a test function φ in D(U) is denoted using angle brackets thus: In mathematics it can be shown that any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. In many cases, these two spaces are isomorphic which means that the distinction between their elements is not always apparent. ...

The function f : U → R is called locally integrable if it is Lebesgue integrable over every compact subset K of U. This is a large class of functions which includes all continuous functions. The topology on D(U) is defined in such a fashion that any locally integrable function f yields a continuous linear functional on D(U) whose value on the test function φ is given by the Lebesgue integral ∫_U fφ dx. Two locally integrable functions f and g yield the same element of D'(U) if and only if they are equal almost everywhere. Similarly, every Radon measure μ on U (which includes the probability distributions) defines an element of D'(U) whose value on the test function φ is ∫φ dμ. The integral of a positive function can be interpreted as the area under a curve. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ...

As mentioned above, integration by parts suggests that the derivative ∂S/∂x_k of the distribution S in the direction x_k should be defined using the formula

for all test functions φ. In this way, every distribution is infinitely differentiable, and the derivative in the direction x_k is a linear operator on D′(U). In general, if α = (α₁, ..., α_n) is an arbitrary multi-index and ∂^α denotes the associated mixed partial derivative operator, the mixed partial derivative ∂^αS of the distribution S ∈ D′(U) is defined by In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

The space D'(U) is turned into a locally convex topological vector space by defining that the sequence (S_k) converges towards 0 if and only if S_k(φ) → 0 for all test functions φ; this topology is called the weak-* topology. This is the case if and only if S_k converges uniformly to 0 on all bounded subsets of D(U). (A subset of E of D(U) is bounded if there exists a compact subset K of U and numbers d_n such that every φ in E has its support in K and has its n-th derivatives bounded by d_n.) With respect to this topology, differentiation of distributions is a continuous operator; this is an important and desirable property that is not shared by most other notions of differentiation. Furthermore, the test functions (which can themselves be viewed as distributions) are dense in D'(U) with respect to this topology. In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... In mathematics, the term dense has at least three different meanings. ...

If ψ : U → R is an infinitely often differentiable function and S is a distribution on U, we define the product Sψ by (Sψ)(φ) = S(ψφ) for all test functions φ. The ordinary product rule of calculus remains valid.

Distributions as derivatives of continuous functions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the algebraic dual of D(U). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. (The precise theorem is below.) In other words, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

One precise version of the theorem is the following.^[1] Let S be a distribution on U. Then for every multi-index α, there exists a continuous function g_α such that any compact subset K of U intersects the supports of only finitely many g_α, and such that The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

Compact support and convolution

We say that a distribution S has compact support if there is a compact subset K of U such that for every test function φ whose support is completely outside of K, we have S(φ) = 0. Alternatively, one may define distributions with compact support as continuous linear functionals on the space C^∞(U); the topology on C^∞(U) is defined such that φ_k converges to 0 if and only if all derivatives of φ_k converge uniformly to 0 on every compact subset of U.

If both S and T are distributions on Rⁿ and one of them has compact support, then one can define a new distribution, the convolution S ∗ T of S and T, as follows: if φ is a test function in D(Rⁿ) and x, y elements of Rⁿ, write φ_x(y) = φ (x + y), ψ(x) = T(φ_x) and (S ∗ T) (φ) = S(ψ). This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

This definition of convolution remains valid under less restrictive assumptions about S and T. ^[2]^[3]

Tempered distributions and Fourier transform

By using a larger space of test functions, one can define the tempered distributions, a subspace of D'(Rⁿ). These distributions are useful if one studies the Fourier transform in generality: all tempered distributions have a Fourier transform, but not all distributions have one. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

The space of test functions employed here, the so-called Schwartz space, is the space of all infinitely differentiable rapidly decreasing functions, where φ : Rⁿ → R is called rapidly decreasing if any derivative of φ, multiplied with any power of |x|, converges towards 0 for |x| → ∞. These functions form a complete topological vector space with a suitably defined family of seminorms. More precisely, let In mathematics, Schwartz space is the function space of rapidly decreasing functions. ... In mathematics, a function on a normed vector space is said to vanish at infinity if as . ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ...

for α, β multi-indices of size n. Then φ is rapidly-decreasing if all the values The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

The family of seminorms p_{α, β} defines a locally convex topology on the Schwartz-space. It is metrizable and complete. In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... A metrizable space is a topological space that is homeomorphic to a metric space. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

The space of tempered distributions is defined as the dual of the Schwartz space. In other words, a distribution F is a tempered distribution if and only if In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ...

for all multi-indices α, β implies The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. All locally integrable functions f with at most polynomial growth, i.e. such that f(x)=O(|x|^r) for some r, are tempered distributions. This includes all functions in L^p(Rⁿ) for p≥1. In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform F yields then an automorphism of Schwartz-space, and we can define the Fourier transform of the tempered distribution S by (FS)(φ) = S(Fφ) for every test function φ. FS is thus again a tempered distribution. The Fourier transform is a continuous, linear, bijective operator from the space of tempered distributions to itself. This operation is compatible with differentiation in the sense that In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

and also with convolution: if S is a tempered distribution and ψ is a slowly increasing infinitely differentiable function on Rⁿ (meaning that all derivatives of ψ grow at most as fast as polynomials), then Sψ is again a tempered distribution and In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

Using holomorphic functions as test functions

The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular by Mikio Sato, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals. In mathematics, hyperfunctions are sums of boundary values of holomorphic functions, and can be thought of informally as distributions of infinite order. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Mikio Sato (佐藤幹夫, born April 18, 1928) is a Japanese mathematician, working in what he calls algebraic analysis. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... This article or section is in need of attention from an expert on the subject. ...

Problem of multiplication

A possible limitation of the theory of distributions (and hyperfunctions) is that it is a purely linear theory, in the sense that the product of two distributions cannot consistently be defined (in general), as has been proved by Laurent Schwartz in the 1950s. For example, if 1/x is the distribution obtained by extending the corresponding function to a homogeneous distribution, and δ is the Dirac delta distribution then Laurent Schwartz (5 March 1915 â€“ 4 July 2002 in Paris) was a French mathematician. ...

so the product of a distribution by a smooth function (which is always well defined) cannot be extended to an associative product on the space of distributions. In mathematics, associativity is a property that a binary operation can have. ...

Thus, nonlinear problems cannot be posed in general and thus not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of "divergencies". Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) causal perturbation theory. This does not solve the problem in other situations. Many other interesting theories are non linear, like for example Navier-Stokes equations of fluid dynamics. Quantum field theory (QFT) is the quantum theory of fields. ... Vladimir Jurko Glaser (April 21, 1924 - January 22, 1984) was a theoretical physicist working on quantum field theory and the canonization of the analytic S-matrix. ... Causal perturbation theory is a mathematically rigorous approach to renormalization theory, which makes it possible to put the theoretical setup of perturbative quantum field theory on a sound mathematical basis. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ...

In view of this, several not entirely satisfactory theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today. In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In mathematics, generalized functions are objects generalizing the notion of functions. ... In mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions, in which multiplication is not problematic. ...

Another example of the imposibility of multiplication is given by convolution theory since $(2 pi)D^{m} delta(u) = int_{-infty}^{infty}dx e^{iux}(ix)^{m}$ then we would have $(2 pi)^{2}i^{m+n}D^{m} delta(u)D^{n}delta(u)= int_{-infty}^{infty}dx int_{-infty}^{infty}dt (x-t)^{m} t^{n}e^{-iux}$ , however the last integral is divergent for every value of 'u'

A simple 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 distributions as shown by H. Kleinert and A. Chervyakov.^[4] The result is equivalent to what can be derived from dimensional regularization.^[5] This article or section is in need of attention from an expert on the subject. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Erwin SchrÃ¶dinger, as depicted on the former Austrian 1000 Schilling bank note. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Hagen Kleinert, Photo taken in 2006 Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany, and Honorary Member of the Russian Academy of Creative Endeavors. ... In theoretical physics, dimensional regularization is a particular way to get rid of infinities that occur when one evaluates Feynman diagrams in quantum field theory. ...

References

^ Walter Rudin, Functional Analysis (second edition), McGraw-Hill, 1991, ISBN 0-07-054236-8.
^ I.M. Gel'fand and G.E. Shilov, Generalized Functions, v. 1, Academic Press, 1964, pp. 103--104.
^ J.J. Benedetto, Harmonic Analysis and Applications, CRC Press, 1997, Definition 2.5.8.
^ H. Kleinert and A. Chervyakov (2001). "Rules for integrals over products of distributions from coordinate independence of path integrals". Europ. Phys. J. C 19: 743--747. doi:10.1007/s100520100600.
^ H. Kleinert and A. Chervyakov (2000). "Coordinate Independence of Quantum-Mechanical Path Integrals". Phys. Lett. A 269: 63. doi:10.1016/S0375-9601(00)00475-8 .

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