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.

## Contents

* 1 Change of variables.
* 2 Examples.
* 3 Products.
* 4 Examples.
* 5 Differentiation.
* 6 Examples.
* 7 Direct products.
* 8 Examples.
* 9 Convolution.
* 10 Examples.
* 11 Fourier transformation.
* 12 Examples.
* 13 Laplace transformation.
* 14 Example.
* 14.1 References
* 14.2 Comments
* 14.3 References

### Change of variables.[edit]

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.[edit]

4) .

5) .

6) , .

7) .

### Products.[edit]

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.[edit]

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.[edit]

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.[edit]

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.[edit]

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.[edit]

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.[edit]

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.[edit]

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.[edit]

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.[edit]

24) ; in particular, .

25) ; in particular, .

26) .

Cf. also Fourier transform of a generalized function.

### Laplace transformation.[edit]

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.[edit]

27) ; in particular,

#### References[edit]

[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[edit]

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[edit]

[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