Test function

In mathematics, test functions are certain types of functions that play an essential role in distribution theory . Test functions of a certain type are usually combined to form a vector space . The corresponding distributions are then linear functionals on these vector spaces. Its name comes from the fact that the distributions (in the sense of linear mapping ) are applied to the test functions and thereby tested .

There are different types of test functions. In the mathematical literature, the space of smooth functions with compact support or the Schwartz space are often referred to as test function space.

Test functions play an important role in functional analysis , for example when introducing the concept of the weak derivative , and in the theory of differential equations . Their origins lie in physics and engineering (more on this in the article Distribution (mathematics) ).

Smooth functions with compact carrier

definition

One of the most common examples of a test function room is crowd

${\displaystyle C_{c}^{\infty }(\Omega )=\{\phi \in C^{\infty }(\Omega )\,|\,\operatorname {supp} \,(\phi )\mathrm {~ist~kompakte~Teilmenge~von~} \Omega \},}$

The graph of a test function in two variables

The function φ _b for b = 1

thus the space of all infinitely often differentiable functions that have a compact support , that is, outside a compact set, are equal to zero.

In order to preserve the space for the test functions , a topology is also defined in this function space. This topology is obtained from a concept of convergence that is defined in this space. A sequence of functions with converges against if there is a compact word with for all and ${\displaystyle (\phi _{j})_{j\in \mathbb {N} }}$ ${\displaystyle \phi _{j}\in C_{c}^{\infty }(\Omega )}$ ${\displaystyle \ \phi }$ ${\displaystyle K\subset \Omega }$ ${\displaystyle \operatorname {supp} (\phi _{j})\subset K}$ ${\displaystyle j}$

${\displaystyle \lim _{j\rightarrow \infty }\sup _{x\in K}\left|{\frac {\partial ^{\alpha }}{\partial x^{\alpha }}}\left(\phi _{j}(x)-\phi (x)\right)\right|=0}$

applies to all multi-indices . ${\displaystyle \alpha \in \mathbb {N} ^{n}}$

Space , along with this concept of convergence, is often noted in the literature . ${\displaystyle C_{c}^{\infty }(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )}$

Examples

An example of a test function with a compact carrier is ${\displaystyle [-b,b]}$

${\displaystyle \phi _{b}(x):={\begin{cases}\exp {\frac {b^{2}}{x^{2}-b^{2}}}&|x|<b\\0&|x|\geq b.\end{cases}}}$

Another example is the family of functions with carrier ( ) ${\displaystyle {\mathcal {C}}^{\infty }}$ ${\displaystyle [0,r]}$ ${\displaystyle r>0}$

${\displaystyle g_{r}(x):=f(x)\cdot f(r-x),{\text{ wobei }}f(x):={\begin{cases}0&x\leq 0\\\exp \left(-{\frac {1}{x}}\right)&x>0\end{cases}}}$

Plots of ${\displaystyle g_{r}}$

properties

Any derivatives of are also in . This is due to the property and the fact that the bearer of a function contains the bearer of its derivative. ${\displaystyle \phi \in C_{c}^{\infty }(\Omega )}$ ${\displaystyle C_{c}^{\infty }(\Omega )}$ ${\displaystyle \phi \in C^{\infty }(\Omega )}$

Let be an open subset of . ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

• Then the test function space is a locally convex vector space , more precisely a (LF) space .
• The test function space fulfills the Heine-Borel property . ${\displaystyle {\mathcal {D}}(\Omega )}$
• The room is a sub -room of the Schwartz room . It is even close to the Schwartz room and is therefore also close to for . ${\displaystyle {\mathcal {D}}(\mathbb {R} ^{n})}$ ${\displaystyle L^{p}(\mathbb {R} ^{n})}$ ${\displaystyle 1\leq p<\infty }$

Schwartz room

→ Main article : Schwartz space

Another space, often referred to as the test function space, is the space of rapidly falling functions, also known as the space of Schwartz test functions or Schwartz space . Its dual space is called the space of tempered distributions and is also noted. ${\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}$

Smooth functions room

→ Main article : Smooth function

The space of the smooth functions on together with their locally convex topology , which is given by the family of semi-norms ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle f\in C^{\infty }(D)\mapsto \sum _{|\alpha |=m}\sup _{x\in K}\left|{\frac {\partial ^{\alpha }}{\partial x^{\alpha }}}f(x)\right|}$

is also used as a test function space. This space is noted with . Its dual space is the space of distributions with a compact carrier . ${\displaystyle {\mathcal {E}}(\mathbb {R} ^{n})}$ ${\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}$

Sobolev rooms

→ Main article : Sobolev room

The Sobolev space for any real number can also be understood as a test function space. This subspace of is also a Hilbert space . Regarding the dual pairing , however, is the corresponding distribution space. ${\displaystyle H^{k}(\mathbb {R} ^{n})}$ ${\displaystyle k>0}$ ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle \textstyle (u,v):=\int _{\mathbb {R} ^{n}}u(x){\overline {v(x)}}\mathrm {d} x}$ ${\displaystyle H^{-k}(\mathbb {R} ^{n})\subset {\mathcal {S}}'(\mathbb {R} ^{n})}$

Riesz-Markov's theorem

With the help of Riesz-Markow's theorem , the dual space of the space of continuous functions can be written on a compact domain as ${\displaystyle K}$

${\displaystyle (C(K))^{\prime }\cong M(K),}$

where the space is the regular Borel measures. The isomorphism is given by the fact that a functional is always in the form ${\displaystyle M(K)}$ ${\displaystyle I:C(K)\rightarrow \mathbb {C} ,}$

${\displaystyle I(f)=\int f(x)d\mu (x),\quad \mu \in M(K),}$

can be written. The integral notation suggests that it is also possible to practice distribution theory for these two spaces.

More general test function rooms

In principle, the concept of test functions and distributions can be transferred to other examples in which one has a function space and its dual space available. The basic idea is to consider a vector space of functions. Since one often wants to fall back on terms such as continuity and convergence , the vector space should be a topological vector space or, better still, a locally convex space . The distributions that belong to the space are then elements of the topological dual space . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}^{\prime }}$

With the help of the dual pairing one can apply a distribution to a test function in the form ${\displaystyle T\in {\mathcal {D}}^{\prime }}$ ${\displaystyle f\in {\mathcal {D}}}$

${\displaystyle T(f)=\langle f,T\rangle }$

write. The notation is very reminiscent of a scalar product , and in fact one often thinks of the -scalar product , so that one (formally) also ${\displaystyle L^{2}}$

${\displaystyle T(f)=\langle f,T\rangle =\int f(x)T(x)dx}$

writes (note that there is no function and therefore the integral is not always well-defined). In order for this interpretation to make sense, it is generally required that the space is a continuously embedded subspace of a vector space of integrable functions, e.g. B. or . ${\displaystyle T}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle L^{1}(\mathbb {R} ^{n}),L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle L_{lok}^{1}(\mathbb {R} ^{n})}$

literature

• Hui-Hsiung Kuo: White Noise Distribution Theory , CRC Press, 1996, ISBN 0-8493-8077-4 .

Individual evidence

1. ↑ Dirk Werner: Functional Analysis . Springer-Verlag, Berlin 2000, ISBN 3-540-21381-3 , pp. 426 .
2. ↑ Man Wah Wong: An introduction to pseudo-differential operator . World Scientific, River Edge, NJ 1999, ISBN 978-981-02-3813-1 , pp. 10-11 .
3. ^ Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( Basic Teachings of Mathematical Sciences 256), pp. 44–45.