Generalized function

A mathematical concept generalizing the classical concept of a function. The need for such a generalization arises in many problems in engineering, physics and mathematics. The concept of a generalized function makes it possible to express in a mathematically-correct form such idealized concepts as the density of a material point, a point charge or a point dipole, the (space) density of a simple or double layer, the intensity of an instantaneous source, etc. On the other hand, the concept of a generalized function reflects the fact that in reality a physical quantity cannot be measured at a point; only its mean values over sufficiently small neighbourhoods of a given point can be measured. Thus, the technique of generalized functions serves as a convenient and adequate apparatus for describing the distributions of various physical quantities. Hence generalized functions are also called distributions.

Generalized functions were first introduced at the end of the 1920-s by P.A.M. Dirac (see [1]) in his research on quantum mechanics, in which he made systematic use of the concept of the -function and its derivatives (see Delta-function). The foundations of the mathematical theory of generalized functions were laid by S.L. Sobolev [2] in 1936 by solving the Cauchy problem for hyperbolic equations, while in the 1950-s L. Schwartz (see [3]) gave a systematic account of the theory of generalized functions and indicated many applications. The theory was then intensively developed by many mathematicians and theoretical physicists, mainly in connection with the needs of theoretical and mathematical physics and the theory of differential equations (see [4]–[7]). The theory of generalized functions has made great advances, has numerous applications, and is extensively used in mathematics, physics and engineering.

Formally, a generalized function is defined as a continuous linear functional on some vector space of sufficiently "good" (test) functions ; . An important example of a test space is the space — the collection of -functions on an open set , with compact support in , endowed with the topology of the strong inductive limit (union) of the spaces , , compact, . The space is the collection of -functions with support in , with the topology given by the countable set of norms

An example of a test function in is the "cap functioncap" :

The space of generalized functions is the space dual to ; , . Convergence of a sequence of generalized functions in is defined as weak convergence of functionals in , that is, , as , in means that , as , for all .

For a linear functional on to be a generalized function in , that is, , it is necessary and sufficient that for any open set there exist numbers and such that

(1)

If the integer in (1) can be chosen independently of , then the generalized function has finite order; the least such is called the order of in . Thus, by (1), every generalized function has finite order in any relatively compact .

The space is complete: If a sequence of generalized functions , in is such that for any the sequence of numbers converges, then the functional

belongs to .

The simplest examples of generalized functions are those generated by locally integrable functions on :

(2)

Generalized functions definable by (2) in terms of locally integrable functions on are called regular generalized functions on ; the remaining generalized functions are called singular. There is a one-to-one correspondence between locally integrable functions on and regular generalized functions on . In this sense, the "ordinary" , that is, locally integrable on , functions are (regular) generalized functions in .

An example of a singular generalized function on is the Dirac -function

It describes the density of a unit mass concentrated at the point . The "cap" (weakly) approximates the -function:

Let and let be a "cap" . Then the function

in is called the regularization of , and , as , in . Moreover, each in is the weak limit of functions in . The latter property is sometimes taken as the starting point for the definition of a generalized function; together with the theorem on the completeness of the space of generalized functions it leads to an equivalent definition of generalized functions [8].

In general, a generalized function need not have a value at an individual point. Nonetheless, one speaks of a generalized function coinciding with a locally integrable function on an open set: A generalized function coincides on with a locally integrable function on if its restriction to is , that is, in accordance with (2), if

for all . One then says that , . In particular, with one obtains a definition of the vanishing of a generalized function in . The set of points of with the property that does not vanish on any neighbourhood of is called the support of , denoted by (cf. also Support of a generalized function). If and is relatively compact, then is called of compact support in .

The following theorem on piecewise glueing generalized functions holds: Suppose that for each a generalized function in is given, where is a neighbourhood of , so that the elements are compatible, that is, in ; then there exists a generalized function in that coincides with in for all .

Examples of generalized functions.

1) The Dirac -function: .

2) The generalized function , defined by

is called the finite part, or principal value, of the integral of ; . The distribution is singular on , but on the open set it is regular and coincides with .

3) The surface -function. Let be a piecewise-smooth surface and let be a continuous function on . The generalized function is defined by

Here for , and is a singular function. This generalized function describes the space density of masses or charges concentrated on with surface density (density of a simple layer).

Linear operations on generalized functions are introduced as extensions of the corresponding operations on the test functions.

Change of variables.

Let and let be a linear transformation of onto . The generalized function in is defined by

(3)

Since the operation is an isomorphism of onto , the operation is an isomorphism of onto . In particular, if , , ( is a similarity (with a reflection if )), then

if ( is a shift by ), then

Formula (3) enables one to define generalized functions that are translation invariant, spherically symmetric, centrally symmetric, homogeneous, periodic, Lorentz invariant, etc.

Let the function have only simple zeros on the line . The function is defined by

Examples.

4) .

5) .

6) , .

7) .

Products.

Let and . The product is defined by

It turns out that , and for ordinary integrable functions coincides with the usual multiplication of the functions and (cf. also Generalized functions, product of).

Examples.

8) .

9) .

However, this product operation cannot be extended to arbitrary generalized functions in such a way that it is associative and commutative. In fact, if this could be done, then one obtains a contradiction:

Such a product can be defined for certain classes of generalized functions, but it may fail to be uniquely defined.

Differentiation.

Let . The generalized (weak) derivative of ,

of order is defined by

(4)

Since the operation is linear and continuous from into , the functional defined by the right-hand side of (4) is a generalized function in . If , then for all with .

The following properties hold: the operation is linear and continuous from into , and any generalized function in is infinitely differentiable (in the generalized sense); the derivative does not depend on the order of differentiation; the Leibniz formula is valid for the differentiation of a product , where ; differentiation does not enlarge the support; for any open set , every generalized function in is a derivative of a continuous function in ; any differential equation , , with constant coefficients can be solved in , if is a convex domain; any generalized function of order with support at the point can be uniquely represented in the form

Examples.

10) , where is the Heaviside function (jump function):

11) ; describes the charge density of a dipole of moment at the point , oriented along the positive -axis.

12) The normal derivative of the density of a simple layer on an orientable surface is a generalization of :

The generalized function describes the space charge density corresponding to a distribution of dipoles on with moment surface density and oriented along a given direction of the normal to (density of a double layer).

13) The general solution of the equation in the class is , where is an arbitrary constant.

14) The general solution of the equation in the class is .

15) , .

16) The trigonometric series

converges in ; it can be differentiated term by term infinitely many times in .

17) .

Cf. also Generalized function, derivative of a.

Direct products.

Let and . Their direct product is defined by the formula

(5)

Since the operation is linear and continuous from into , the functional , defined by (5), is a generalized function in . The direct product is a commutative and associative operation, and

A generalized function in does not depend on if it can be represented in the form

in this case one writes .

Examples.

18) .

19) The general solution in of the equation for the vibration of a homogeneous string, , is given by

where and are arbitrary generalized functions in .

Convolution.

Let and be generalized functions in with the property that their direct product can be extended to functions of the form , where runs through , in the following sense: For every sequence of functions in with the properties

(on any compact set), the sequence of numbers has a limit independent of the sequence . This limit is called the convolution of and , and is denoted by . Thus,

(6)

The completeness of implies that . As elementary examples show, the convolution does not exist for all pairs and . It does exists if one of the generalized functions is of compact support. If the convolution exists in , then it is commutative, , and the following formulas for the differentiation of a convolution are valid:

(7)

Also

(8)

hence, from (7),

Finally

The example

shows that convolution is a non-associative operation. However, associative (and commutative) convolution algebras exist. By (8), the -function is the identity element in them. For example, a convolution algebra is formed by the set consisting of the generalized functions in with support in a convex acute closed cone with vertex at . One writes:

A generalized function in is called a fundamental solution (point-source function) of a differential operator with constant coefficients if it satisfies the equation

If a fundamental solution of is known, then a solution can be constructed for the equation for those in for which the convolution exists, and this solution is given by .

Examples.

20) The kernel of a fractional differentiation or integration operator , :

Here , , , , an integer. If , then is the primitive of order for (derivative of order for ).

21) ,

22) ,

23) ,

Fourier transformation.

It is defined on the class of generalized functions of slow growth. The space of test functions consists of the -functions that decrease at infinity together with all their derivatives faster than any power of . The topology of is given by the countable set of norms

Here and , and these imbeddings are continuous. Functions of slow growth that are locally integrable on are in , and define regular functionals on by formula (2).

Every generalized function in is a derivative of a continuous function of slow growth, and so has finite order on .

The Fourier transform of a generalized function in is defined by the equation

where

is the classical Fourier transform. Since the operation is an isomorphism of onto , the operation is an isomorphism of onto , and the inverse of is given by

The following basic formulas hold for :

if has compact support. If the generalized function is periodic with -period , , then , and it can be expanded in a trigonometric series

converging to in . Here

Examples.

24) ; in particular, .

25) ; in particular, .

26) .

Cf. also Fourier transform of a generalized function.

Laplace transformation.

Let the generalized function , where is a closed convex acute cone. Let , where is the cone dual to . The Laplace transform of is defined by

(9)

The mapping defines an isomorphism of the convolution algebra onto the algebra consisting of the functions that are holomorphic in the wedge and that satisfy the following growth condition: There exist numbers and such that for any cone (i.e. ) there exists a number such that

The inverse of the Laplace transform is given by the equation

(10)

where the right-hand side of (10) is independent of .

The one-to-one correspondence between and given by equations (9) and (10) can be conveniently represented by the following scheme:

in which is called the transform of , and the spectral function of .

Every in the algebra has a boundary value as , , in , related to the spectral function of by the formula according to (9). The following basic formulas hold for the Laplace transform:

Example.

27) ; in particular,

References

[1]	P.A.M. Dirac, "The principles of quantum mechanics" , Clarendon Press (1947) MR0023198 Zbl 0030.04801
[2]	S.L. Sobolev, "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales" Mat. Sb. , 1 (1936) pp. 39–72
[3]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951) MR2067351 MR0209834 MR0117544 MR0107812 MR0041345 MR0035918 MR0032815 MR0031106 MR0025615 Zbl 0962.46025 Zbl 0653.46037 Zbl 0399.46028 Zbl 0149.09501 Zbl 0085.09703 Zbl 0089.09801 Zbl 0089.09601 Zbl 0078.11003 Zbl 0042.11405 Zbl 0037.07301 Zbl 0039.33201 Zbl 0030.12601
[4]	N.N. Bogolyubov, A.A. Logunov, I.T. Todorov, "Introduction to axiomatic quantum field theory" , Benjamin (1975) (Translated from Russian) MR452277
[5]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1–5 , Acad. Press (1966–1968) (Translated from Russian) Zbl 0801.33020 Zbl 0699.33012 Zbl 0159.18301 Zbl 0355.46017 Zbl 0144.17202 Zbl 0115.33101 Zbl 0108.29601
[6]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101
[7]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian) MR0564116 MR0549767 Zbl 0515.46034 Zbl 0515.46033
[8]	P. Antosik, J. Mikusiński, R. Sikorski, "Theory of distributions. The sequential approach" , Elsevier (1973) MR0365130 Zbl 0267.46028

Comments

The notation means that the closure is contained in . Usually the support of a function (or distribution) is defined as the closure of the set of points where it is non-zero.

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 MR0617913 Zbl 0435.46002
[a2]	D.S. Jones, "The theory of generalized functions" , Cambridge Univ. Press (1982)
[a3]	W. Rudin, "Functional analysis" , McGraw-Hill (1974) MR1157815 MR0458106 MR0365062 Zbl 0867.46001 Zbl 0253.46001
[a4]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001