Distribution (mathematics): Difference between revisions

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imported>Aleksander Stos
m (more precise)
imported>Aleksander Stos
m (consistent look and feel)
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converges to zero (this is continuity of ''f'').  
converges to zero (this is continuity of ''f'').  
Equivalently, the functional ''f'' on ''K'' is continuous if for any compact set <math>A\subset \mathbb{R}^n</math> there exist <math>c\in\mathbb{R}</math> and <math>k\in\mathbb{N}</math> such that for any test function <math>\phi</math> with support in ''A''  
Equivalently, the functional ''f'' on ''K'' is continuous if for any compact set <math>\scriptstyle A\subset \mathbb{R}^n</math> there exist <math>\scriptstyle c\in\mathbb{R}</math> and <math>\scriptstyle k\in\mathbb{N}</math> such that for any test function <math>\scriptstyle\phi</math> with support in ''A''  
:<math> (f,\phi) \le c \sup_{|\alpha| \le k,\;x\in A} |D^{\alpha} \phi(x)|.</math>
:<math> (f,\phi) \le c \sup_{|\alpha| \le k,\;x\in A} |D^{\alpha} \phi(x)|.</math>
Here <math>\alpha</math> denotes a [[multi-index]] and <math>D^\alpha</math> is usual [[partial derivative]] described by <math>\alpha.</math>
Here <math>\scriptstyle\alpha</math> denotes a [[multi-index]] and <math>\scriptstyle D^\alpha</math> is usual [[partial derivative]] described by <math>\scriptstyle\alpha.</math>


A distribution (generalized function) is defined as any linear continuous functional on K.
A distribution (generalized function) is defined as any linear continuous functional on K.

Revision as of 13:55, 1 December 2007

Distributions (also known as generalized functions) are mathematical objects that are important in physics and engineering where many non-continuous problems are formulated in terms of distributions. An important example is the Dirac delta function. Dirac's intuitive ideas were placed on firm mathematical footing by S. L. Sobolev in 1936, who studied the uniqueness of solutions of the Cauchy problem for linear hyperbolic equations. In 1950 Laurent Schwartz published his Théorie des Distributions. In this book he systematizes the theory of generalized functions unifying all earlier approaches and extending the results.

The physicist's definition of the Dirac delta function

is recognized by the mathematician as a linear functional acting on a "well-behaved" test function φ(x).

In order to generalize this, the concept of "test functions" is needed. Let the set K consist of all real functions φ(x) with continuous derivatives of all orders and bounded support. This means that the function φ(x) vanishes outside some bounded region, which may differ for different functions. The set K is the space of test functions. It can be shown that K is a linear space.

Secondly, the concept of linear functional is needed. We call f a continuous linear functional on K, if f maps all elements of K onto a real number such that

(i). For any two real numbers and and any two functions in K, and , we have linearity

(ii). If the sequence converges uniformly to zero in K, then

converges to zero (this is continuity of f). Equivalently, the functional f on K is continuous if for any compact set there exist and such that for any test function with support in A

Here denotes a multi-index and is usual partial derivative described by

A distribution (generalized function) is defined as any linear continuous functional on K.