Test functions are auxiliary functions used in mathematical analysis to probe other functions, distributions, differential equations, or variational identities. They are usually chosen from a class of functions with enough regularity, decay, or boundary behavior to justify operations such as integration by parts, localization, and passage to weak limits.
Common spaces of test functions
Compactly supported smooth functions Let U be an open subset of Rn. With minor modifications, one can replace Rn by any (paracompact) smooth manifold. The space D(U) of test functions on U is defined as follows. A function φ {\displaystyle \varphi } : U → R is said to have compact support if there exists a compact subset K of U such that φ {\displaystyle \varphi } (x) = 0 for all x in U \ K. The elements of D(U) are the infinitely differentiable functions φ {\displaystyle \varphi } : U → R with compact support. This is a real vector space. It can be given a topology by defining the limit of a sequence of elements of D(U). A sequence ( φ {\displaystyle \varphi } k) in D(U) is said to converge to φ {\displaystyle \varphi } ∈ D(U) if the following two conditions hold:
There is a compact set K ⊂ U containing the supports of all φ {\displaystyle \varphi } k:
⋃ k supp ( φ k ) ⊂ K . {\displaystyle \bigcup \nolimits _{k}\operatorname {supp} (\varphi _{k})\subset K.}
For each multi-index α, the sequence of partial derivatives ∂ α φ k {\displaystyle \partial ^{\alpha }\varphi _{k}} tends uniformly to ∂ α φ {\displaystyle \partial ^{\alpha }\varphi } . With this definition, D(U) becomes a complete locally convex topological vector space. Now let U be the union of Ui where {Ui} is a countable nested family of open subsets of U with compact closures Ki = Ui. Then we have the countable increasing union
D ( U ) = ⋃ i D K i {\displaystyle \mathrm {D} (U)=\bigcup \nolimits _{i}\mathrm {D} _{K_{i}}}
where DKi is the set of all smooth functions on U with support lying in Ki. On each DKi, consider the topology given by the seminorms
‖ φ ‖ α = max x ∈ K i | ∂ α φ | , {\displaystyle \|\varphi \|_{\alpha }=\max _{x\in K_{i}}\left|\partial ^{\alpha }\varphi \right|,}
i.e. the topology of uniform convergence of derivatives of arbitrary order. This makes each DKi a Fréchet space. The resulting LF space structure on D(U) is the topology described above.
Schwartz functions
The Schwartz space S(Rn) is the function space of all infinitely differentiable functions that are rapidly decreasing at infinity along with all partial derivatives. Thus φ : Rn → R is in the Schwartz space provided that any derivative of φ {\displaystyle \varphi } , multiplied with any power of |x|, converges towards 0 for |x| → ∞. These functions form a complete topological vector space with a suitably defined family of seminorms. More precisely, let
p α , β ( φ ) = sup x ∈ R n | x α D β φ ( x ) | {\displaystyle p_{\alpha ,\beta }(\varphi )=\sup _{x\in \mathbf {R} ^{n}}\left|x^{\alpha }D^{\beta }\varphi (x)\right|}
for α, β multi-indices of size n. Then φ {\displaystyle \varphi } is a Schwartz function if all the values satisfy
p α , β ( φ ) < ∞ . {\displaystyle p_{\alpha ,\beta }(\varphi )<\infty .}
The family of seminorms pα, β defines a locally convex topology on the Schwartz space. When n is equal to 1, the seminorms are, in fact, norms on the Schwartz space. Otherwise, one can define a norm on S(Rn) via
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