Tempered distribution

A tempered distribution is an object from the distribution theory, a mathematical branch of functional analysis. A tempered distribution is a special case of a distribution. Laurent Schwartz led in 1947 to the space of tempered distributions one to integrate the Fourier transform in its distribution theory can.

Schwartz space

In order to define tempered distributions, the space of functions quickly falling is first explained. Quick decreasing functions are infinitely differentiable and strive at infinity as fast to zero, that they and all their derivatives are falling faster than any polynomial function. The amount of all these functions is also known as Schwartz's space and is characterized by

Defined. Due to the semi-norms

Is the Schwartz space to a metrizable space lokalkonvexem. The peculiarity of this area is that the Fourier transform is an automorphism thereon. In addition, the room is included in all Sobolev spaces. The space of test functions can be continuously embedded in the Schwartz space and is situated in this densely.

Definition

A tempered distribution is a continuous linear functional on the Schwartz space, ie a continuous linear map. Since the amount of the tempered distributions form defined according to the dual space of topological, this area is recorded using. Because of this duality, one also speaks of the slow-growing distributions in contrast to the rapidly decreasing functions.

Examples

• Delta distribution
• Dirac comb
• All distributions that are generated by a polynomial function, are tempered distributions. Therefore a polynomial, then the continuous functional a tempered distribution. These distributions are in contrast to the delta function, respectively, in contrast to the Dirac comb regular distributions.

Gelfandsches Raumtripel

The Schwartz space is dense in the Hilbert space of square integrable functions. For this reason applies to their dual spaces and the inclusion of the set of Riesz -Fischer follows This leads to a total of inclusion

The continuous embedding is the normal identification of a function with a distribution. That is, the picture

The pair is an example of an extended Hilbert space, or the triple an example of a gelfandsches Raumtripel (after Israel Gelfand ). In all three areas, the Fourier transform is an automorphism.

The functions of the continuous component of the eigenvalue spectrum of an operator on are generally not square integrable, but can be considered as an element of. Further details can be found in Volume III of the specified below literature books by Gelfand. In the application to quantum mechanics, this means that the space ( in the standard representation of this plane waves ), for example, eigenfunctions of the local or momentum operator contains, which are not included because the integral diverges on its magnitude squared.

Fourier transformation

Definition

Be a tempered distribution, the Fourier transform is defined for all by

In this context, the Fourier transform is defined by features on. There is also a different convention for the Fourier transformation with the prefactor. This is not used in this article.

Properties

It equips the amount of the weak -* topology. Then, the Fourier transform is a constant, bijective mapping. The Fourier preimage of is calculated with the formula

Example

• Be the delta distribution and to the point. Then for the Fourier transform. So corresponds to the distribution of generated. In the case therefore corresponds to the distribution of generated. If, in the Fourier transform nor the prefactor then the result of the example is the distribution that is generated by.

• Let now the distribution generated by the constant one function. The obvious approach fails to calculate the expression, because it leads to a non- absolutely convergent integral. To solve the above example and you need a little trick. It is.

Fourier Laplace transform

In this section, the Fourier transform is considered only for distributions with compact support. Since the Fourier transform in this context has special characteristics, then they are called Fourier -Laplace transform. So be a distribution with compact support. Then, the Laplace transformation is by Fourier

Defined. This is well defined, because one can show that a function which analytically for all even - so completely - is. Furthermore, this definition is consistent with the above definition, if the distributions have compact support. What all functions can occur as a Fourier -Laplace transformations here, characterizes the set of Paley -Wiener.

Laplace transform

For tempered distributions, one can also define a Laplace transform. This looks similar to the Fourier - Laplace transform of the previous section. Be a tempered distribution with support in, then the Laplace transform is carried

Defined. The result of the transformation is also again an entire function. In contrast to the Fourier Laplace transform the Laplace transform is defined for tempered distributions that have no compact support. This is possible because of the decay better than that of the Fourier kernel.