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 : U → R is said to have compact support if there exists a compact subset K of U(x) = 0 for all x in U \ K. The elements of D(U : 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(Uk) in D(U∈D(U) if the following two conditions hold:

• There is a compact set K⊂Uk:

  For each multi-index tends uniformly.

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 = i. Then we have the countable increasing union

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

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 , 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

for α, β multi-indices of size n is a Schwartz function if all the values satisfy

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

1. for k ≥ 1.

The Schwartz space is metrizable and complete. Because the Fourier transform changes differentiation by xα into multiplication by xα and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

Sobolev test spaces

In the theory of weak formulations and weak solutions of partial differential equations, the term test function, and then pass to a larger Sobolev space by density or completion. In this setting, a test function is usually an admissible function from the space in which the weak identity is required to hold.

. are often used as test functions in variational formulations.

A typical example is the weak form of the Poisson equation pointwise, one asks that

after completion. corresponds to homogeneous Dirichlet boundary conditions in the trace sense on sufficiently regular domains.

This use of the term should be distinguished from the distribution-theoretic convention, where the test functions themselves are usually smooth. In weak formulations, the word test emphasizes the role of the function in probing an equation or variational identity, rather than membership in a fixed smooth test-function space.

Test functions on manifolds

Test functions can also be defined on smooth manifolds with compact support; on a non-compact manifold, the compact-support condition is a genuine restriction.

is used for localization, integration by parts, and the definition of distributions. More intrinsically, distributions on a manifold may be defined as continuous linear functionals on compactly supported smooth densities, or, after choosing a smooth positive density or a Riemannian volume form.

is a smooth vector bundle. This is useful, for example, when defining weak or distributional sections of the dual bundle, differential forms with distributional coefficients, or weak formulations of equations for vector-valued unknowns.

on open subsets of Euclidean space. On each compact subset, convergence means uniform convergence of all derivatives in local coordinates; globally, the usual topology is an inductive limit over compact subsets. Partitions of unity and smooth cutoff functions allow many local constructions with Euclidean test functions to be transferred to manifolds.

Use in weak formulations

In the study of partial differential equations, test functions are used to pass from a differential equation in pointwise form (called the classical form) to a weak formulation. The usual procedure is to multiply the equation by a test function, integrate over the domain, and use integration by parts to transfer derivatives from the unknown function onto the test function. This allows the formulation of equations for functions that may not possess all derivatives appearing in the original equation in the classical sense.

is an open set and consider the Poisson equation

and integrating gives

, gives

.

When boundary conditions are included, the test space is often chosen to encode them. For the homogeneous Dirichlet problem such that

. In this formulation, the test functions need not themselves be smooth: they can to the Sobolev space obtained as a completion of compactly supported smooth functions in the relevant Sobolev norm.

Test functions also appear in variational leads to the Euler–Lagrange equation in weak form.

Use in distribution theory

Topologies

See also

• Bump function

• Schwartz space

• Distribution (mathematics)

• Weak derivative

• Weak solution

• Sobolev space

• Mollifier

References

Sources

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