The test functions are called in mathematics certain types of functions, which play an essential role in distribution theory. Usually one summarizes test functions of a particular type to a vector space together. The corresponding distributions are then linear functionals on these vector spaces. Your name stems from the fact that ( in the sense of linear maps ) applies the distributions on the test functions and thereby tested.
There are various types of test functions. In the mathematical literature, the space of functions with smooth compact support or the Schwartz room are often referred to as a test function space.
Test functions play an important role in functional analysis, such as the introduction of the notion of weak dissipation, as well as in the theory of differential equations. Its origins lie in the physics and the engineering sciences (more on this in the article distribution (mathematics) ).
Smooth functions with compact support
One of the most common examples of a test function space is the set
Ie the space of all infinitely differentiable functions having compact support, that is, outside of a compact set equal to zero.
Notion of convergence
To obtain the space of test functions, a topology is defined on this function space yet. This topology is obtained from a converging term which is defined on the space. A sequence of functions with converges to if there is a compact set with for all and
Applies to all multi- indices.
The room, with this notion of convergence is often quoted in the literature.
An example of a test function with compact support
Be an open subset of.
- Then the test function space is a locally convex vector space, or more precisely an (LF )-space.
- The test function space satisfies the Heine- Borel property.
- The space is a subspace of the Schwartz space. He is even dense in the Schwartz space and is therefore dense in for.
Another area that is often referred to as test function space is the space of functions quickly falling, also known as the space of test functions or Schwartz schwartz between space. Its dual space is called a space of tempered distributions, and is quoted at.
And the Sobolev space of an arbitrary real number may be regarded as a test function space. This subspace is also a Hilbert space. Regarding the dual pairing, however, the corresponding distribution space.
The Riesz - Markov
Using the representation theorem of Riesz - Markov can write on a compact domain of the dual space of the space of continuous functions as
The space of regular Borel measures is. The isomorphism is given by the fact that a functional always in the form
Can be written. The integral notation suggests that it is possible for these two rooms to operate distribution theory.
More general test function spaces
Basically, can the concept of test functions and distributions applied to other examples where one has a function space and its dual space available. The basic idea is that one considers a vector space of functions. Since you want to frequently use terms such as continuity and convergence, the vector space should be a topological vector space, or better yet a locally convex space. The distributions that belong to the space are then elements of the topological dual space.
With the help of the dual pairing can be applying a distribution to a test function in the form
. Write The notation is strongly reminiscent of a scalar product, and often in fact thinks in this connection on the scalar product, so that you (formal) also
Writes ( Note that no function is the integral and therefore is not always well defined ). Thus, this interpretation makes sense, one requires a rule that it is an ever- embedded subspace of a vector space of integrable functions, for example, or.